statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
cyclotomic'_two
(R : Type*) [comm_ring R] [is_domain R] (p : ℕ) [char_p R p] (hp : p ≠ 2) :
cyclotomic' 2 R = X + 1 | begin
rw [cyclotomic'],
have prim_root_two : primitive_roots 2 R = {(-1 : R)},
{ simp only [finset.eq_singleton_iff_unique_mem, mem_primitive_roots two_pos],
exact ⟨is_primitive_root.neg_one p hp, λ x, is_primitive_root.eq_neg_one_of_two_right⟩ },
simp only [prim_root_two, finset.prod_singleton, ring_hom.ma... | lemma | polynomial.cyclotomic'_two | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"char_p",
"comm_ring",
"finset.eq_singleton_iff_unique_mem",
"finset.prod_singleton",
"is_domain",
"mem_primitive_roots",
"primitive_roots",
"ring_hom.map_neg",
"ring_hom.map_one"
] | The second modified cyclotomic polyomial is `X + 1` if the characteristic of `R` is not `2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic'.monic
(n : ℕ) (R : Type*) [comm_ring R] [is_domain R] : (cyclotomic' n R).monic | monic_prod_of_monic _ _ $ λ z hz, monic_X_sub_C _ | lemma | polynomial.cyclotomic'.monic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"is_domain"
] | `cyclotomic' n R` is monic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic'_ne_zero
(n : ℕ) (R : Type*) [comm_ring R] [is_domain R] : cyclotomic' n R ≠ 0 | (cyclotomic'.monic n R).ne_zero | lemma | polynomial.cyclotomic'_ne_zero | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"is_domain",
"ne_zero"
] | `cyclotomic' n R` is different from `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_degree_cyclotomic' {ζ : R} {n : ℕ} (h : is_primitive_root ζ n) :
(cyclotomic' n R).nat_degree = nat.totient n | begin
rw [cyclotomic'],
rw nat_degree_prod (primitive_roots n R) (λ (z : R), (X - C z)),
simp only [is_primitive_root.card_primitive_roots h, mul_one,
nat_degree_X_sub_C,
nat.cast_id, finset.sum_const, nsmul_eq_mul],
intros z hz,
exact X_sub_C_ne_zero z
end | lemma | polynomial.nat_degree_cyclotomic' | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"is_primitive_root",
"is_primitive_root.card_primitive_roots",
"mul_one",
"nat.cast_id",
"nat.totient",
"nsmul_eq_mul",
"primitive_roots"
] | The natural degree of `cyclotomic' n R` is `totient n` if there is a primitive root of
unity in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
degree_cyclotomic' {ζ : R} {n : ℕ} (h : is_primitive_root ζ n) :
(cyclotomic' n R).degree = nat.totient n | by simp only [degree_eq_nat_degree (cyclotomic'_ne_zero n R), nat_degree_cyclotomic' h] | lemma | polynomial.degree_cyclotomic' | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"is_primitive_root",
"nat.totient"
] | The degree of `cyclotomic' n R` is `totient n` if there is a primitive root of unity in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
roots_of_cyclotomic (n : ℕ) (R : Type*) [comm_ring R] [is_domain R] :
(cyclotomic' n R).roots = (primitive_roots n R).val | by { rw cyclotomic', exact roots_prod_X_sub_C (primitive_roots n R) } | lemma | polynomial.roots_of_cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"is_domain",
"primitive_roots"
] | The roots of `cyclotomic' n R` are the primitive `n`-th roots of unity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : is_primitive_root ζ n) :
X ^ n - 1 = ∏ ζ in nth_roots_finset n R, (X - C ζ) | begin
rw [nth_roots_finset, ← multiset.to_finset_eq (is_primitive_root.nth_roots_nodup h)],
simp only [finset.prod_mk, ring_hom.map_one],
rw [nth_roots],
have hmonic : (X ^ n - C (1 : R)).monic := monic_X_pow_sub_C (1 : R) (ne_of_lt hpos).symm,
symmetry,
apply prod_multiset_X_sub_C_of_monic_of_roots_card_eq... | lemma | polynomial.X_pow_sub_one_eq_prod | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"finset.prod_mk",
"is_primitive_root",
"is_primitive_root.card_nth_roots",
"is_primitive_root.nth_roots_nodup",
"multiset.to_finset_eq",
"ring_hom.map_one"
] | If there is a primitive `n`th root of unity in `K`, then `X ^ n - 1 = ∏ (X - μ)`, where `μ`
varies over the `n`-th roots of unity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic'_splits (n : ℕ) : splits (ring_hom.id K) (cyclotomic' n K) | begin
apply splits_prod (ring_hom.id K),
intros z hz,
simp only [splits_X_sub_C (ring_hom.id K)]
end | lemma | polynomial.cyclotomic'_splits | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"ring_hom.id"
] | `cyclotomic' n K` splits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_pow_sub_one_splits {ζ : K} {n : ℕ} (h : is_primitive_root ζ n) :
splits (ring_hom.id K) (X ^ n - C (1 : K)) | by rw [splits_iff_card_roots, ← nth_roots, is_primitive_root.card_nth_roots h,
nat_degree_X_pow_sub_C] | lemma | polynomial.X_pow_sub_one_splits | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"is_primitive_root",
"is_primitive_root.card_nth_roots",
"ring_hom.id"
] | If there is a primitive `n`-th root of unity in `K`, then `X ^ n - 1`splits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_cyclotomic'_eq_X_pow_sub_one {K : Type*} [comm_ring K] [is_domain K] {ζ : K} {n : ℕ}
(hpos : 0 < n) (h : is_primitive_root ζ n) : ∏ i in nat.divisors n, cyclotomic' i K = X ^ n - 1 | have hd : (n.divisors : set ℕ).pairwise_disjoint (λ k, primitive_roots k K),
from λ x hx y hy hne, is_primitive_root.disjoint hne,
by simp only [X_pow_sub_one_eq_prod hpos h, cyclotomic', ← finset.prod_bUnion hd,
h.nth_roots_one_eq_bUnion_primitive_roots] | lemma | polynomial.prod_cyclotomic'_eq_X_pow_sub_one | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"finset.prod_bUnion",
"is_domain",
"is_primitive_root",
"is_primitive_root.disjoint",
"nat.divisors",
"primitive_roots"
] | If there is a primitive `n`-th root of unity in `K`, then
`∏ i in nat.divisors n, cyclotomic' i K = X ^ n - 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic'_eq_X_pow_sub_one_div {K : Type*} [comm_ring K] [is_domain K] {ζ : K} {n : ℕ}
(hpos : 0 < n) (h : is_primitive_root ζ n) :
cyclotomic' n K = (X ^ n - 1) /ₘ (∏ i in nat.proper_divisors n, cyclotomic' i K) | begin
rw [←prod_cyclotomic'_eq_X_pow_sub_one hpos h, ← nat.cons_self_proper_divisors hpos.ne',
finset.prod_cons],
have prod_monic : (∏ i in nat.proper_divisors n, cyclotomic' i K).monic,
{ apply monic_prod_of_monic,
intros i hi,
exact cyclotomic'.monic i K },
rw (div_mod_by_monic_unique (cyclotomic'... | lemma | polynomial.cyclotomic'_eq_X_pow_sub_one_div | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"bot_lt_iff_ne_bot",
"comm_ring",
"finset.prod_cons",
"is_domain",
"is_primitive_root",
"mul_comm",
"nat.cons_self_proper_divisors",
"nat.proper_divisors"
] | If there is a primitive `n`-th root of unity in `K`, then
`cyclotomic' n K = (X ^ k - 1) /ₘ (∏ i in nat.proper_divisors k, cyclotomic' i K)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_coeff_of_cyclotomic' {K : Type*} [comm_ring K] [is_domain K] {ζ : K} {n : ℕ}
(h : is_primitive_root ζ n) :
(∃ (P : ℤ[X]), map (int.cast_ring_hom K) P = cyclotomic' n K ∧
P.degree = (cyclotomic' n K).degree ∧ P.monic) | begin
refine lifts_and_degree_eq_and_monic _ (cyclotomic'.monic n K),
induction n using nat.strong_induction_on with k ihk generalizing ζ h,
rcases k.eq_zero_or_pos with rfl|hpos,
{ use 1,
simp only [cyclotomic'_zero, coe_map_ring_hom, polynomial.map_one] },
let B : K[X] := ∏ i in nat.proper_divisors k, c... | lemma | polynomial.int_coeff_of_cyclotomic' | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"bot_lt_iff_ne_bot",
"comm_ring",
"finset.prod_cons",
"int.cast_ring_hom",
"is_domain",
"is_primitive_root",
"mul_comm",
"nat.cons_self_proper_divisors",
"nat.proper_divisors",
"polynomial.map_one",
"ring_hom.mem_srange",
"subsemiring.prod_mem"
] | If there is a primitive `n`-th root of unity in `K`, then `cyclotomic' n K` comes from a
monic polynomial with integer coefficients. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_int_coeff_of_cycl {K : Type*} [comm_ring K] [is_domain K] [char_zero K] {ζ : K}
{n : ℕ+} (h : is_primitive_root ζ n) :
(∃! (P : ℤ[X]), map (int.cast_ring_hom K) P = cyclotomic' n K) | begin
obtain ⟨P, hP⟩ := int_coeff_of_cyclotomic' h,
refine ⟨P, hP.1, λ Q hQ, _⟩,
apply (map_injective (int.cast_ring_hom K) int.cast_injective),
rw [hP.1, hQ]
end | lemma | polynomial.unique_int_coeff_of_cycl | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"char_zero",
"comm_ring",
"int.cast_injective",
"int.cast_ring_hom",
"is_domain",
"is_primitive_root"
] | If `K` is of characteristic `0` and there is a primitive `n`-th root of unity in `K`,
then `cyclotomic n K` comes from a unique polynomial with integer coefficients. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic (n : ℕ) (R : Type*) [ring R] : R[X] | if h : n = 0 then 1 else
map (int.cast_ring_hom R) ((int_coeff_of_cyclotomic' (complex.is_primitive_root_exp n h)).some) | def | polynomial.cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"complex.is_primitive_root_exp",
"int.cast_ring_hom",
"ring"
] | The `n`-th cyclotomic polynomial with coefficients in `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_cyclotomic_rw {n : ℕ} (h : n ≠ 0) :
cyclotomic n ℤ = (int_coeff_of_cyclotomic' (complex.is_primitive_root_exp n h)).some | begin
simp only [cyclotomic, h, dif_neg, not_false_iff],
ext i,
simp only [coeff_map, int.cast_id, eq_int_cast]
end | lemma | polynomial.int_cyclotomic_rw | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"complex.is_primitive_root_exp",
"eq_int_cast",
"int.cast_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_cyclotomic_int (n : ℕ) (R : Type*) [ring R] :
map (int.cast_ring_hom R) (cyclotomic n ℤ) = cyclotomic n R | begin
by_cases hzero : n = 0,
{ simp only [hzero, cyclotomic, dif_pos, polynomial.map_one] },
simp only [cyclotomic, int_cyclotomic_rw, hzero, ne.def, dif_neg, not_false_iff]
end | lemma | polynomial.map_cyclotomic_int | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"int.cast_ring_hom",
"polynomial.map_one",
"ring"
] | `cyclotomic n R` comes from `cyclotomic n ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_cyclotomic_spec (n : ℕ) : map (int.cast_ring_hom ℂ) (cyclotomic n ℤ) = cyclotomic' n ℂ ∧
(cyclotomic n ℤ).degree = (cyclotomic' n ℂ).degree ∧ (cyclotomic n ℤ).monic | begin
by_cases hzero : n = 0,
{ simp only [hzero, cyclotomic, degree_one, monic_one, cyclotomic'_zero, dif_pos,
eq_self_iff_true, polynomial.map_one, and_self] },
rw int_cyclotomic_rw hzero,
exact (int_coeff_of_cyclotomic' (complex.is_primitive_root_exp n hzero)).some_spec
end | lemma | polynomial.int_cyclotomic_spec | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"complex.is_primitive_root_exp",
"int.cast_ring_hom",
"polynomial.map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cyclotomic_unique {n : ℕ} {P : ℤ[X]} (h : map (int.cast_ring_hom ℂ) P =
cyclotomic' n ℂ) : P = cyclotomic n ℤ | begin
apply map_injective (int.cast_ring_hom ℂ) int.cast_injective,
rw [h, (int_cyclotomic_spec n).1]
end | lemma | polynomial.int_cyclotomic_unique | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"int.cast_injective",
"int.cast_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_cyclotomic (n : ℕ) {R S : Type*} [ring R] [ring S] (f : R →+* S) :
map f (cyclotomic n R) = cyclotomic n S | begin
rw [←map_cyclotomic_int n R, ←map_cyclotomic_int n S, map_map],
congr
end | lemma | polynomial.map_cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"ring"
] | The definition of `cyclotomic n R` commutes with any ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic.eval_apply {R S : Type*} (q : R) (n : ℕ) [ring R] [ring S] (f : R →+* S) :
eval (f q) (cyclotomic n S) = f (eval q (cyclotomic n R)) | by rw [← map_cyclotomic n f, eval_map, eval₂_at_apply] | lemma | polynomial.cyclotomic.eval_apply | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_zero (R : Type*) [ring R] : cyclotomic 0 R = 1 | by simp only [cyclotomic, dif_pos] | lemma | polynomial.cyclotomic_zero | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"ring"
] | The zeroth cyclotomic polyomial is `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_one (R : Type*) [ring R] : cyclotomic 1 R = X - 1 | begin
have hspec : map (int.cast_ring_hom ℂ) (X - 1) = cyclotomic' 1 ℂ,
{ simp only [cyclotomic'_one, pnat.one_coe, map_X, polynomial.map_one, polynomial.map_sub] },
symmetry,
rw [←map_cyclotomic_int, ←(int_cyclotomic_unique hspec)],
simp only [map_X, polynomial.map_one, polynomial.map_sub]
end | lemma | polynomial.cyclotomic_one | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"int.cast_ring_hom",
"pnat.one_coe",
"polynomial.map_one",
"polynomial.map_sub",
"ring"
] | The first cyclotomic polyomial is `X - 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic.monic (n : ℕ) (R : Type*) [ring R] : (cyclotomic n R).monic | begin
rw ←map_cyclotomic_int,
exact (int_cyclotomic_spec n).2.2.map _,
end | lemma | polynomial.cyclotomic.monic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"ring"
] | `cyclotomic n` is monic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic.is_primitive (n : ℕ) (R : Type*) [comm_ring R] : (cyclotomic n R).is_primitive | (cyclotomic.monic n R).is_primitive | lemma | polynomial.cyclotomic.is_primitive | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring"
] | `cyclotomic n` is primitive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_ne_zero (n : ℕ) (R : Type*) [ring R] [nontrivial R] : cyclotomic n R ≠ 0 | (cyclotomic.monic n R).ne_zero | lemma | polynomial.cyclotomic_ne_zero | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"ne_zero",
"nontrivial",
"ring"
] | `cyclotomic n R` is different from `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
degree_cyclotomic (n : ℕ) (R : Type*) [ring R] [nontrivial R] :
(cyclotomic n R).degree = nat.totient n | begin
rw ←map_cyclotomic_int,
rw degree_map_eq_of_leading_coeff_ne_zero (int.cast_ring_hom R) _,
{ cases n with k,
{ simp only [cyclotomic, degree_one, dif_pos, nat.totient_zero, with_top.coe_zero]},
rw [←degree_cyclotomic' (complex.is_primitive_root_exp k.succ (nat.succ_ne_zero k))],
exact (int_c... | lemma | polynomial.degree_cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"complex.is_primitive_root_exp",
"eq_int_cast",
"int.cast_one",
"int.cast_ring_hom",
"nat.totient",
"nat.totient_zero",
"nontrivial",
"one_ne_zero",
"ring"
] | The degree of `cyclotomic n` is `totient n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_degree_cyclotomic (n : ℕ) (R : Type*) [ring R] [nontrivial R] :
(cyclotomic n R).nat_degree = nat.totient n | by rw [nat_degree, degree_cyclotomic, with_bot.unbot'_coe] | lemma | polynomial.nat_degree_cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"nat.totient",
"nontrivial",
"ring",
"with_bot.unbot'_coe"
] | The natural degree of `cyclotomic n` is `totient n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
degree_cyclotomic_pos (n : ℕ) (R : Type*) (hpos : 0 < n) [ring R] [nontrivial R] :
0 < (cyclotomic n R).degree | by
{ rw degree_cyclotomic n R, exact_mod_cast (nat.totient_pos hpos) } | lemma | polynomial.degree_cyclotomic_pos | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"nat.totient_pos",
"nontrivial",
"ring"
] | The degree of `cyclotomic n R` is positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_cyclotomic_eq_X_pow_sub_one {n : ℕ} (hpos : 0 < n) (R : Type*) [comm_ring R] :
∏ i in nat.divisors n, cyclotomic i R = X ^ n - 1 | begin
have integer : ∏ i in nat.divisors n, cyclotomic i ℤ = X ^ n - 1,
{ apply map_injective (int.cast_ring_hom ℂ) int.cast_injective,
simp only [polynomial.map_prod, int_cyclotomic_spec, polynomial.map_pow, map_X,
polynomial.map_one, polynomial.map_sub],
exact prod_cyclotomic'_eq_X_pow_sub_... | lemma | polynomial.prod_cyclotomic_eq_X_pow_sub_one | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"complex.is_primitive_root_exp",
"int.cast_injective",
"int.cast_ring_hom",
"nat.divisors",
"polynomial.map_X",
"polynomial.map_one",
"polynomial.map_pow",
"polynomial.map_prod",
"polynomial.map_sub"
] | `∏ i in nat.divisors n, cyclotomic i R = X ^ n - 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic.dvd_X_pow_sub_one (n : ℕ) (R : Type*) [ring R] :
(cyclotomic n R) ∣ X ^ n - 1 | begin
suffices : cyclotomic n ℤ ∣ X ^ n - 1,
{ simpa only [map_cyclotomic_int, polynomial.map_sub, polynomial.map_one, polynomial.map_pow,
polynomial.map_X] using map_dvd (int.cast_ring_hom R) this },
rcases n.eq_zero_or_pos with rfl | hn,
{ simp },
rw [← prod_cyclotomic_eq_X_pow_sub_one hn],
exact fi... | lemma | polynomial.cyclotomic.dvd_X_pow_sub_one | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"finset.dvd_prod_of_mem",
"int.cast_ring_hom",
"map_dvd",
"polynomial.map_X",
"polynomial.map_one",
"polynomial.map_pow",
"polynomial.map_sub",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_cyclotomic_eq_geom_sum {n : ℕ} (h : 0 < n) (R) [comm_ring R] :
∏ i in n.divisors.erase 1, cyclotomic i R = ∑ i in finset.range n, X ^ i | suffices ∏ i in n.divisors.erase 1, cyclotomic i ℤ = ∑ i in finset.range n, X ^ i,
by simpa only [polynomial.map_prod, map_cyclotomic_int, polynomial.map_sum, polynomial.map_pow,
polynomial.map_X] using congr_arg (map (int.cast_ring_hom R)) this,
by rw [← mul_left_inj' (cyclotomic_ne_zero 1 ℤ), prod_erase_mul _ _ (na... | lemma | polynomial.prod_cyclotomic_eq_geom_sum | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"finset.range",
"geom_sum_mul",
"int.cast_ring_hom",
"mul_left_inj'",
"polynomial.map_X",
"polynomial.map_pow",
"polynomial.map_prod",
"polynomial.map_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_prime (R : Type*) [ring R] (p : ℕ) [hp : fact p.prime] :
cyclotomic p R = ∑ i in finset.range p, X ^ i | begin
suffices : cyclotomic p ℤ = ∑ i in range p, X ^ i,
{ simpa only [map_cyclotomic_int, polynomial.map_sum, polynomial.map_pow, polynomial.map_X]
using congr_arg (map (int.cast_ring_hom R)) this },
rw [← prod_cyclotomic_eq_geom_sum hp.out.pos, hp.out.divisors,
erase_insert (mem_singleton.not.2 hp.out... | lemma | polynomial.cyclotomic_prime | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"fact",
"finset.range",
"int.cast_ring_hom",
"polynomial.map_X",
"polynomial.map_pow",
"polynomial.map_sum",
"ring"
] | If `p` is prime, then `cyclotomic p R = ∑ i in range p, X ^ i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_prime_mul_X_sub_one (R : Type*) [ring R] (p : ℕ) [hn : fact (nat.prime p)] :
(cyclotomic p R) * (X - 1) = X ^ p - 1 | by rw [cyclotomic_prime, geom_sum_mul] | lemma | polynomial.cyclotomic_prime_mul_X_sub_one | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"fact",
"geom_sum_mul",
"nat.prime",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_two (R : Type*) [ring R] : cyclotomic 2 R = X + 1 | by simp [cyclotomic_prime] | lemma | polynomial.cyclotomic_two | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_three (R : Type*) [ring R] : cyclotomic 3 R = X ^ 2 + X + 1 | by simp [cyclotomic_prime, sum_range_succ'] | lemma | polynomial.cyclotomic_three | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_dvd_geom_sum_of_dvd (R) [ring R] {d n : ℕ} (hdn : d ∣ n)
(hd : d ≠ 1) : cyclotomic d R ∣ ∑ i in finset.range n, X ^ i | begin
suffices : cyclotomic d ℤ ∣ ∑ i in finset.range n, X ^ i,
{ simpa only [map_cyclotomic_int, polynomial.map_sum, polynomial.map_pow, polynomial.map_X]
using map_dvd (int.cast_ring_hom R) this },
rcases n.eq_zero_or_pos with rfl | hn,
{ simp },
rw ←prod_cyclotomic_eq_geom_sum hn,
apply finset.dvd_... | lemma | polynomial.cyclotomic_dvd_geom_sum_of_dvd | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"finset.dvd_prod_of_mem",
"finset.range",
"int.cast_ring_hom",
"map_dvd",
"polynomial.map_X",
"polynomial.map_pow",
"polynomial.map_sum",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd (R) [comm_ring R] {d n : ℕ}
(h : d ∈ n.proper_divisors) :
(X ^ d - 1) * ∏ x in n.divisors \ d.divisors, cyclotomic x R = X ^ n - 1 | begin
obtain ⟨hd, hdn⟩ := nat.mem_proper_divisors.mp h,
have h0n : 0 < n := pos_of_gt hdn,
have h0d : 0 < d := nat.pos_of_dvd_of_pos hd h0n,
rw [←prod_cyclotomic_eq_X_pow_sub_one h0d, ←prod_cyclotomic_eq_X_pow_sub_one h0n,
mul_comm, finset.prod_sdiff (nat.divisors_subset_of_dvd h0n.ne' hd)]
end | lemma | polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"finset.prod_sdiff",
"mul_comm",
"nat.divisors_subset_of_dvd",
"pos_of_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd (R) [comm_ring R] {d n : ℕ}
(h : d ∈ n.proper_divisors) : (X ^ d - 1) * cyclotomic n R ∣ X ^ n - 1 | begin
have hdn := (nat.mem_proper_divisors.mp h).2,
use ∏ x in n.proper_divisors \ d.divisors, cyclotomic x R,
symmetry,
convert X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd R h using 1,
rw mul_assoc,
congr' 1,
rw [← nat.insert_self_proper_divisors hdn.ne_bot, insert_sdiff_of_not_mem, prod_in... | lemma | polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"finset.not_mem_sdiff_of_not_mem_left",
"mul_assoc",
"nat.divisor_le",
"nat.insert_self_proper_divisors",
"nat.proper_divisors.not_self_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_eq_prod_X_pow_sub_one_pow_moebius {n : ℕ} (R : Type*) [comm_ring R] [is_domain R] :
algebra_map _ (ratfunc R) (cyclotomic n R) =
∏ i in n.divisors_antidiagonal, (algebra_map R[X] _ (X ^ i.snd - 1)) ^ μ i.fst | begin
rcases n.eq_zero_or_pos with rfl | hpos,
{ simp },
have h : ∀ (n : ℕ), 0 < n →
∏ i in nat.divisors n, algebra_map _ (ratfunc R) (cyclotomic i R) = algebra_map _ _ (X ^ n - 1),
{ intros n hn,
rw [← prod_cyclotomic_eq_X_pow_sub_one hn R, ring_hom.map_prod] },
rw (prod_eq_iff_prod_pow_moebius_eq_of... | lemma | polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"algebra_map",
"comm_ring",
"is_domain",
"is_fraction_ring.to_map_eq_zero_iff",
"nat.divisors",
"ratfunc",
"ring_hom.map_prod"
] | `cyclotomic n R` can be expressed as a product in a fraction field of `R[X]`
using Möbius inversion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_eq_X_pow_sub_one_div {R : Type*} [comm_ring R] {n : ℕ}
(hpos: 0 < n) : cyclotomic n R = (X ^ n - 1) /ₘ (∏ i in nat.proper_divisors n, cyclotomic i R) | begin
nontriviality R,
rw [←prod_cyclotomic_eq_X_pow_sub_one hpos, ← nat.cons_self_proper_divisors hpos.ne',
finset.prod_cons],
have prod_monic : (∏ i in nat.proper_divisors n, cyclotomic i R).monic,
{ apply monic_prod_of_monic,
intros i hi,
exact cyclotomic.monic i R },
rw (div_mod_by_monic_uniqu... | lemma | polynomial.cyclotomic_eq_X_pow_sub_one_div | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"bot_lt_iff_ne_bot",
"comm_ring",
"finset.prod_cons",
"mul_comm",
"nat.cons_self_proper_divisors",
"nat.proper_divisors"
] | We have
`cyclotomic n R = (X ^ k - 1) /ₘ (∏ i in nat.proper_divisors k, cyclotomic i K)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_pow_sub_one_dvd_prod_cyclotomic (R : Type*) [comm_ring R] {n m : ℕ} (hpos : 0 < n)
(hm : m ∣ n) (hdiff : m ≠ n) : X ^ m - 1 ∣ ∏ i in nat.proper_divisors n, cyclotomic i R | begin
replace hm := nat.mem_proper_divisors.2 ⟨hm, lt_of_le_of_ne (nat.divisor_le (nat.mem_divisors.2
⟨hm, hpos.ne'⟩)) hdiff⟩,
rw [← finset.sdiff_union_of_subset (nat.divisors_subset_proper_divisors (ne_of_lt hpos).symm
(nat.mem_proper_divisors.1 hm).1 (ne_of_lt (nat.mem_proper_divisors.1 hm).2)),
finse... | lemma | polynomial.X_pow_sub_one_dvd_prod_cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"finset.prod_union",
"finset.sdiff_disjoint",
"finset.sdiff_union_of_subset",
"nat.divisor_le",
"nat.divisors_subset_proper_divisors",
"nat.pos_of_mem_proper_divisors",
"nat.proper_divisors"
] | If `m` is a proper divisor of `n`, then `X ^ m - 1` divides
`∏ i in nat.proper_divisors n, cyclotomic i R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_eq_prod_X_sub_primitive_roots {K : Type*} [comm_ring K] [is_domain K] {ζ : K}
{n : ℕ} (hz : is_primitive_root ζ n) :
cyclotomic n K = ∏ μ in primitive_roots n K, (X - C μ) | begin
rw ←cyclotomic',
induction n using nat.strong_induction_on with k hk generalizing ζ hz,
obtain hzero | hpos := k.eq_zero_or_pos,
{ simp only [hzero, cyclotomic'_zero, cyclotomic_zero] },
have h : ∀ i ∈ k.proper_divisors, cyclotomic i K = cyclotomic' i K,
{ intros i hi,
obtain ⟨d, hd⟩ := (nat.mem_p... | lemma | polynomial.cyclotomic_eq_prod_X_sub_primitive_roots | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"finset.prod_congr",
"is_domain",
"is_primitive_root",
"is_primitive_root.pow",
"mul_comm",
"primitive_roots"
] | If there is a primitive `n`-th root of unity in `K`, then
`cyclotomic n K = ∏ μ in primitive_roots n R, (X - C μ)`. In particular,
`cyclotomic n K = cyclotomic' n K` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_cyclotomic_iff {R : Type*} [comm_ring R] {n : ℕ} (hpos: 0 < n)
(P : R[X]) :
P = cyclotomic n R ↔ P * (∏ i in nat.proper_divisors n, polynomial.cyclotomic i R) = X ^ n - 1 | begin
nontriviality R,
refine ⟨λ hcycl, _, λ hP, _⟩,
{ rw [hcycl, ← prod_cyclotomic_eq_X_pow_sub_one hpos R, ← nat.cons_self_proper_divisors hpos.ne',
finset.prod_cons] },
{ have prod_monic : (∏ i in nat.proper_divisors n, cyclotomic i R).monic,
{ apply monic_prod_of_monic,
intros i hi,
... | lemma | polynomial.eq_cyclotomic_iff | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"bot_lt_iff_ne_bot",
"comm_ring",
"finset.prod_cons",
"mul_comm",
"nat.cons_self_proper_divisors",
"nat.proper_divisors",
"polynomial.cyclotomic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_prime_pow_eq_geom_sum {R : Type*} [comm_ring R] {p n : ℕ} (hp : p.prime) :
cyclotomic (p ^ (n + 1)) R = ∑ i in finset.range p, (X ^ (p ^ n)) ^ i | begin
have : ∀ m, cyclotomic (p ^ (m + 1)) R = ∑ i in finset.range p, (X ^ (p ^ m)) ^ i ↔
(∑ i in finset.range p, (X ^ (p ^ m)) ^ i) * ∏ (x : ℕ) in finset.range (m + 1),
cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1,
{ intro m,
have := eq_cyclotomic_iff (pow_pos hp.pos (m + 1)) _,
rw eq_comm at this,... | lemma | polynomial.cyclotomic_prime_pow_eq_geom_sum | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"finset.prod_range_succ",
"finset.range",
"geom_sum_mul",
"mul_comm",
"nat.prod_proper_divisors_prime_pow",
"pow_add",
"pow_mul",
"pow_one",
"pow_pos"
] | If `p ^ k` is a prime power, then
`cyclotomic (p ^ (n + 1)) R = ∑ i in range p, (X ^ (p ^ n)) ^ i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_prime_pow_mul_X_pow_sub_one (R : Type*) [comm_ring R] (p k : ℕ)
[hn : fact (nat.prime p)] :
(cyclotomic (p ^ (k + 1)) R) * (X ^ (p ^ k) - 1) = X ^ (p ^ (k + 1)) - 1 | by rw [cyclotomic_prime_pow_eq_geom_sum hn.out, geom_sum_mul, ← pow_mul, pow_succ, mul_comm] | lemma | polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"fact",
"geom_sum_mul",
"mul_comm",
"nat.prime",
"pow_mul",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_coeff_zero (R : Type*) [comm_ring R] {n : ℕ} (hn : 1 < n) :
(cyclotomic n R).coeff 0 = 1 | begin
induction n using nat.strong_induction_on with n hi,
have hprod : (∏ i in nat.proper_divisors n, (polynomial.cyclotomic i R).coeff 0) = -1,
{ rw [←finset.insert_erase (nat.one_mem_proper_divisors_iff_one_lt.2
(lt_of_lt_of_le one_lt_two hn)), finset.prod_insert (finset.not_mem_erase 1 _),
cycloto... | lemma | polynomial.cyclotomic_coeff_zero | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"comm_ring",
"finset.not_mem_erase",
"finset.prod_congr",
"finset.prod_cons",
"finset.prod_const_one",
"finset.prod_insert",
"mul_neg",
"mul_one",
"nat.cons_self_proper_divisors",
"nat.pos_of_mem_proper_divisors",
"nat.proper_divisors",
"ne.le_iff_lt",
"one_lt_two",
"polynomial.cyclotomic"... | The constant term of `cyclotomic n R` is `1` if `2 ≤ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime : fact p.prime] {a : ℕ}
(hroot : is_root (cyclotomic n (zmod p)) (nat.cast_ring_hom (zmod p) a)) :
a.coprime p | begin
apply nat.coprime.symm,
rw [hprime.1.coprime_iff_not_dvd],
intro h,
replace h := (zmod.nat_coe_zmod_eq_zero_iff_dvd a p).2 h,
rw [is_root.def, eq_nat_cast, h, ← coeff_zero_eq_eval_zero] at hroot,
by_cases hone : n = 1,
{ simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq... | lemma | polynomial.coprime_of_root_cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"eq_nat_cast",
"fact",
"nat.cast_ring_hom",
"nat.coprime.symm",
"one_ne_zero",
"zmod",
"zmod.nat_coe_zmod_eq_zero_iff_dvd"
] | If `(a : ℕ)` is a root of `cyclotomic n (zmod p)`, where `p` is a prime, then `a` and `p` are
coprime. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_of_root_cyclotomic_dvd {n : ℕ} (hpos : 0 < n) {p : ℕ} [fact p.prime]
{a : ℕ} (hroot : is_root (cyclotomic n (zmod p)) (nat.cast_ring_hom (zmod p) a)) :
order_of (zmod.unit_of_coprime a (coprime_of_root_cyclotomic hpos hroot)) ∣ n | begin
apply order_of_dvd_of_pow_eq_one,
suffices hpow : eval (nat.cast_ring_hom (zmod p) a) (X ^ n - 1 : (zmod p)[X]) = 0,
{ simp only [eval_X, eval_one, eval_pow, eval_sub, eq_nat_cast] at hpow,
apply units.coe_eq_one.1,
simp only [sub_eq_zero.mp hpow, zmod.coe_unit_of_coprime, units.coe_pow] },
rw [is... | lemma | polynomial.order_of_root_cyclotomic_dvd | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/basic.lean | [
"algebra.ne_zero",
"algebra.polynomial.big_operators",
"ring_theory.roots_of_unity.complex",
"data.polynomial.lifts",
"data.polynomial.splits",
"data.zmod.algebra",
"field_theory.ratfunc",
"field_theory.separable",
"number_theory.arithmetic_function",
"ring_theory.roots_of_unity.basic"
] | [
"eq_nat_cast",
"fact",
"finset.prod_cons",
"nat.cast_ring_hom",
"nat.cons_self_proper_divisors",
"order_of",
"order_of_dvd_of_pow_eq_one",
"units.coe_pow",
"zero_mul",
"zmod",
"zmod.coe_unit_of_coprime",
"zmod.unit_of_coprime"
] | If `(a : ℕ)` is a root of `cyclotomic n (zmod p)`, then the multiplicative order of `a` modulo
`p` divides `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_one_cyclotomic_prime {R : Type*} [comm_ring R] {p : ℕ} [hn : fact p.prime] :
eval 1 (cyclotomic p R) = p | by simp only [cyclotomic_prime, eval_X, one_pow, finset.sum_const, eval_pow,
eval_finset_sum, finset.card_range, smul_one_eq_coe] | lemma | polynomial.eval_one_cyclotomic_prime | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"comm_ring",
"fact",
"finset.card_range",
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval₂_one_cyclotomic_prime {R S : Type*} [comm_ring R] [semiring S] (f : R →+* S)
{p : ℕ} [fact p.prime] : eval₂ f 1 (cyclotomic p R) = p | by simp | lemma | polynomial.eval₂_one_cyclotomic_prime | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"comm_ring",
"fact",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_one_cyclotomic_prime_pow {R : Type*} [comm_ring R] {p : ℕ} (k : ℕ)
[hn : fact p.prime] : eval 1 (cyclotomic (p ^ (k + 1)) R) = p | by simp only [cyclotomic_prime_pow_eq_geom_sum hn.out, eval_X, one_pow, finset.sum_const,
eval_pow, eval_finset_sum, finset.card_range, smul_one_eq_coe] | lemma | polynomial.eval_one_cyclotomic_prime_pow | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"comm_ring",
"fact",
"finset.card_range",
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval₂_one_cyclotomic_prime_pow {R S : Type*} [comm_ring R] [semiring S] (f : R →+* S)
{p : ℕ} (k : ℕ) [fact p.prime] : eval₂ f 1 (cyclotomic (p ^ (k + 1)) R) = p | by simp | lemma | polynomial.eval₂_one_cyclotomic_prime_pow | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"comm_ring",
"fact",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_neg_one_pos {n : ℕ} (hn : 2 < n) {R} [linear_ordered_comm_ring R] :
0 < eval (-1 : R) (cyclotomic n R) | begin
haveI := ne_zero.of_gt hn,
rw [←map_cyclotomic_int, ←int.cast_one, ←int.cast_neg, eval_int_cast_map,
int.coe_cast_ring_hom, int.cast_pos],
suffices : 0 < eval ↑(-1 : ℤ) (cyclotomic n ℝ),
{ rw [←map_cyclotomic_int n ℝ, eval_int_cast_map, int.coe_cast_ring_hom] at this,
exact_mod_cast this },
si... | lemma | polynomial.cyclotomic_neg_one_pos | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"continuous",
"int.cast_neg",
"int.cast_one",
"int.cast_pos",
"int.coe_cast_ring_hom",
"intermediate_value_univ",
"linear_ordered_comm_ring",
"linear_ordered_ring.order_of_le_two",
"ne_zero.of_gt",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_pos {n : ℕ} (hn : 2 < n) {R} [linear_ordered_comm_ring R] (x : R) :
0 < eval x (cyclotomic n R) | begin
induction n using nat.strong_induction_on with n ih,
have hn' : 0 < n := pos_of_gt hn,
have hn'' : 1 < n := one_lt_two.trans hn,
dsimp at ih,
have := prod_cyclotomic_eq_geom_sum hn' R,
apply_fun eval x at this,
rw [← cons_self_proper_divisors hn'.ne', finset.erase_cons_of_ne _ hn''.ne',
finse... | lemma | polynomial.cyclotomic_pos | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"eq_or_ne",
"finset.erase_cons_of_ne",
"finset.mem_erase",
"finset.prod_cons",
"finset.prod_erase_mul",
"finset.prod_nonneg",
"finset.range",
"geom_sum_eq_zero_iff_neg_one",
"geom_sum_neg_iff",
"geom_sum_pos_iff",
"ih",
"linear_ordered_comm_ring",
"mul_nonpos_of_nonneg_of_nonpos",
"nat.two... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_pos_and_nonneg (n : ℕ) {R} [linear_ordered_comm_ring R] (x : R) :
(1 < x → 0 < eval x (cyclotomic n R)) ∧ (1 ≤ x → 0 ≤ eval x (cyclotomic n R)) | begin
rcases n with _ | _ | _ | n;
simp only [cyclotomic_zero, cyclotomic_one, cyclotomic_two, succ_eq_add_one,
eval_X, eval_one, eval_add, eval_sub, sub_nonneg, sub_pos,
zero_lt_one, zero_le_one, implies_true_iff, imp_self, and_self],
{ split; intro; linarith, },
{ have : 2 < n + 3 := dec_trivial,
... | lemma | polynomial.cyclotomic_pos_and_nonneg | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"imp_self",
"linear_ordered_comm_ring",
"zero_le_one",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_pos' (n : ℕ) {R} [linear_ordered_comm_ring R] {x : R} (hx : 1 < x) :
0 < eval x (cyclotomic n R) | (cyclotomic_pos_and_nonneg n x).1 hx | lemma | polynomial.cyclotomic_pos' | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"linear_ordered_comm_ring"
] | Cyclotomic polynomials are always positive on inputs larger than one.
Similar to `cyclotomic_pos` but with the condition on the input rather than index of the
cyclotomic polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_nonneg (n : ℕ) {R} [linear_ordered_comm_ring R] {x : R} (hx : 1 ≤ x) :
0 ≤ eval x (cyclotomic n R) | (cyclotomic_pos_and_nonneg n x).2 hx | lemma | polynomial.cyclotomic_nonneg | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"linear_ordered_comm_ring"
] | Cyclotomic polynomials are always nonnegative on inputs one or more. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_one_cyclotomic_not_prime_pow {R : Type*} [ring R] {n : ℕ}
(h : ∀ {p : ℕ}, p.prime → ∀ k : ℕ, p ^ k ≠ n) : eval 1 (cyclotomic n R) = 1 | begin
rcases n.eq_zero_or_pos with rfl | hn',
{ simp },
have hn : 1 < n := one_lt_iff_ne_zero_and_ne_one.mpr ⟨hn'.ne', (h nat.prime_two 0).symm⟩,
rsuffices (h | h) : eval 1 (cyclotomic n ℤ) = 1 ∨ eval 1 (cyclotomic n ℤ) = -1,
{ have := eval_int_cast_map (int.cast_ring_hom R) (cyclotomic n ℤ) 1,
simpa only... | lemma | polynomial.eval_one_cyclotomic_not_prime_pow | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"dvd_refl",
"eq_int_cast",
"exists_prop",
"finset.card_range",
"finset.dvd_prod_of_mem",
"finset.prod_const",
"finset.prod_image",
"finset.prod_singleton",
"int.cast_one",
"int.cast_ring_hom",
"int.nat_abs_dvd_iff_dvd",
"int.nat_abs_pow",
"mul_assoc",
"mul_comm",
"mul_left_comm",
"nat.... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_one_pow_totient_lt_cyclotomic_eval {n : ℕ} {q : ℝ} (hn' : 2 ≤ n) (hq' : 1 < q) :
(q - 1) ^ totient n < (cyclotomic n ℝ).eval q | begin
have hn : 0 < n := pos_of_gt hn',
have hq := zero_lt_one.trans hq',
have hfor : ∀ ζ' ∈ primitive_roots n ℂ, q - 1 ≤ ‖↑q - ζ'‖,
{ intros ζ' hζ',
rw mem_primitive_roots hn at hζ',
convert norm_sub_norm_le (↑q) ζ',
{ rw [complex.norm_real, real.norm_of_nonneg hq.le], },
{ rw [hζ'.norm'_eq_one... | lemma | polynomial.sub_one_pow_totient_lt_cyclotomic_eval | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"abs_eq_self",
"algebra_map",
"complex.I",
"complex.abs_of_real",
"complex.arg_of_real_of_nonneg",
"complex.card_primitive_roots",
"complex.coe_algebra_map",
"complex.exp",
"complex.is_primitive_root_exp",
"complex.norm_eq_abs",
"complex.norm_real",
"complex.of_real_eq_zero",
"complex.same_r... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_one_pow_totient_le_cyclotomic_eval {q : ℝ} (hq' : 1 < q) :
∀ n, (q - 1) ^ totient n ≤ (cyclotomic n ℝ).eval q | | 0 := by simp only [totient_zero, pow_zero, cyclotomic_zero, eval_one]
| 1 := by simp only [totient_one, pow_one, cyclotomic_one, eval_sub, eval_X, eval_one]
| (n + 2) := (sub_one_pow_totient_lt_cyclotomic_eval dec_trivial hq').le | lemma | polynomial.sub_one_pow_totient_le_cyclotomic_eval | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"pow_one",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_eval_lt_add_one_pow_totient {n : ℕ} {q : ℝ} (hn' : 3 ≤ n) (hq' : 1 < q) :
(cyclotomic n ℝ).eval q < (q + 1) ^ totient n | begin
have hn : 0 < n := pos_of_gt hn',
have hq := zero_lt_one.trans hq',
have hfor : ∀ ζ' ∈ primitive_roots n ℂ, ‖↑q - ζ'‖ ≤ q + 1,
{ intros ζ' hζ',
rw mem_primitive_roots hn at hζ',
convert norm_sub_le (↑q) ζ',
{ rw [complex.norm_real, real.norm_of_nonneg (zero_le_one.trans_lt hq').le], },
{ r... | lemma | polynomial.cyclotomic_eval_lt_add_one_pow_totient | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"abs_of_pos",
"algebra_map",
"complex.I",
"complex.abs",
"complex.arg_eq_zero_iff",
"complex.arg_of_real_of_nonneg",
"complex.card_primitive_roots",
"complex.coe_algebra_map",
"complex.exp",
"complex.is_primitive_root_exp",
"complex.neg_im",
"complex.neg_re",
"complex.norm_eq_abs",
"comple... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_eval_le_add_one_pow_totient {q : ℝ} (hq' : 1 < q) :
∀ n, (cyclotomic n ℝ).eval q ≤ (q + 1) ^ totient n | | 0 := by simp
| 1 := by simp [add_assoc, add_nonneg, zero_le_one]
| 2 := by simp
| (n + 3) := (cyclotomic_eval_lt_add_one_pow_totient dec_trivial hq').le | lemma | polynomial.cyclotomic_eval_le_add_one_pow_totient | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_one_pow_totient_lt_nat_abs_cyclotomic_eval {n : ℕ} {q : ℕ} (hn' : 1 < n) (hq : q ≠ 1) :
(q - 1) ^ totient n < ((cyclotomic n ℤ).eval ↑q).nat_abs | begin
rcases hq.lt_or_lt.imp_left nat.lt_one_iff.mp with rfl | hq',
{ rw [zero_tsub, zero_pow (nat.totient_pos (pos_of_gt hn')), pos_iff_ne_zero, int.nat_abs_ne_zero,
nat.cast_zero, ← coeff_zero_eq_eval_zero, cyclotomic_coeff_zero _ hn'],
exact one_ne_zero },
rw [← @nat.cast_lt ℝ, nat.cast_pow, nat.cast... | lemma | polynomial.sub_one_pow_totient_lt_nat_abs_cyclotomic_eval | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"algebra_map",
"int.cast_nat_abs",
"int.nat_abs_ne_zero",
"le_abs_self",
"nat.cast_lt",
"nat.cast_one",
"nat.cast_pow",
"nat.cast_sub",
"nat.cast_zero",
"nat.totient_pos",
"one_ne_zero",
"pos_of_gt",
"zero_pow",
"zero_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_one_lt_nat_abs_cyclotomic_eval {n : ℕ} {q : ℕ} (hn' : 1 < n) (hq : q ≠ 1) :
q - 1 < ((cyclotomic n ℤ).eval ↑q).nat_abs | calc q - 1 ≤ (q - 1) ^ totient n : nat.le_self_pow (nat.totient_pos $ pos_of_gt hn').ne' _
... < ((cyclotomic n ℤ).eval ↑q).nat_abs : sub_one_pow_totient_lt_nat_abs_cyclotomic_eval hn' hq | lemma | polynomial.sub_one_lt_nat_abs_cyclotomic_eval | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/eval.lean | [
"ring_theory.polynomial.cyclotomic.roots",
"tactic.by_contra",
"topology.algebra.polynomial",
"number_theory.padics.padic_val",
"analysis.complex.arg"
] | [
"nat.le_self_pow",
"nat.totient_pos",
"pos_of_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_expand_eq_cyclotomic_mul {p n : ℕ} (hp : nat.prime p) (hdiv : ¬p ∣ n)
(R : Type*) [comm_ring R] :
expand R p (cyclotomic n R) = (cyclotomic (n * p) R) * (cyclotomic n R) | begin
rcases nat.eq_zero_or_pos n with rfl | hnpos,
{ simp },
haveI := ne_zero.of_pos hnpos,
suffices : expand ℤ p (cyclotomic n ℤ) = (cyclotomic (n * p) ℤ) * (cyclotomic n ℤ),
{ rw [← map_cyclotomic_int, ← map_expand, this, polynomial.map_mul, map_cyclotomic_int] },
refine eq_of_monic_of_dvd_of_nat_degree_... | lemma | polynomial.cyclotomic_expand_eq_cyclotomic_mul | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/expand.lean | [
"ring_theory.polynomial.cyclotomic.roots"
] | [
"algebra_rat",
"comm_ring",
"complex.is_primitive_root_exp",
"dvd_mul_left",
"is_coprime.mul_dvd",
"is_primitive_root.pow_of_dvd",
"is_primitive_root.pow_of_prime",
"minpoly.dvd",
"mul_comm",
"mul_left_cancel₀",
"nat.prime",
"nat.prime.coprime_iff_not_dvd",
"nat.prime.ne_one",
"nat.prime.p... | If `p` is a prime such that `¬ p ∣ n`, then
`expand R p (cyclotomic n R) = (cyclotomic (n * p) R) * (cyclotomic n R)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_expand_eq_cyclotomic {p n : ℕ} (hp : nat.prime p) (hdiv : p ∣ n)
(R : Type*) [comm_ring R] : expand R p (cyclotomic n R) = cyclotomic (n * p) R | begin
rcases n.eq_zero_or_pos with rfl | hzero,
{ simp },
haveI := ne_zero.of_pos hzero,
suffices : expand ℤ p (cyclotomic n ℤ) = cyclotomic (n * p) ℤ,
{ rw [← map_cyclotomic_int, ← map_expand, this, map_cyclotomic_int] },
refine eq_of_monic_of_dvd_of_nat_degree_le (cyclotomic.monic _ _)
((cyclotomic.mo... | lemma | polynomial.cyclotomic_expand_eq_cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/expand.lean | [
"ring_theory.polynomial.cyclotomic.roots"
] | [
"comm_ring",
"complex.is_primitive_root_exp",
"dvd_mul_left",
"is_primitive_root.pow_of_dvd",
"minpoly.is_integrally_closed_dvd",
"mul_comm",
"nat.prime",
"nat.totient_mul_of_prime_of_dvd",
"ne_zero.of_pos"
] | If `p` is a prime such that `p ∣ n`, then
`expand R p (cyclotomic n R) = cyclotomic (p * n) R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_irreducible_pow_of_irreducible_pow {p : ℕ} (hp : nat.prime p)
{R} [comm_ring R] [is_domain R] {n m : ℕ} (hmn : m ≤ n)
(h : irreducible (cyclotomic (p ^ n) R)) : irreducible (cyclotomic (p ^ m) R) | begin
unfreezingI
{ rcases m.eq_zero_or_pos with rfl | hm,
{ simpa using irreducible_X_sub_C (1 : R) },
obtain ⟨k, rfl⟩ := nat.exists_eq_add_of_le hmn,
induction k with k hk },
{ simpa using h },
have : m + k ≠ 0 := (add_pos_of_pos_of_nonneg hm k.zero_le).ne',
rw [nat.add_succ, pow_succ', ←cycloto... | lemma | polynomial.cyclotomic_irreducible_pow_of_irreducible_pow | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/expand.lean | [
"ring_theory.polynomial.cyclotomic.roots"
] | [
"comm_ring",
"dvd_pow_self",
"irreducible",
"is_domain",
"nat.exists_eq_add_of_le",
"nat.prime",
"pow_succ'"
] | If the `p ^ n`th cyclotomic polynomial is irreducible, so is the `p ^ m`th, for `m ≤ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_irreducible_of_irreducible_pow {p : ℕ} (hp : nat.prime p) {R} [comm_ring R]
[is_domain R] {n : ℕ} (hn : n ≠ 0) (h : irreducible (cyclotomic (p ^ n) R)) :
irreducible (cyclotomic p R) | pow_one p ▸ cyclotomic_irreducible_pow_of_irreducible_pow hp hn.bot_lt h | lemma | polynomial.cyclotomic_irreducible_of_irreducible_pow | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/expand.lean | [
"ring_theory.polynomial.cyclotomic.roots"
] | [
"comm_ring",
"irreducible",
"is_domain",
"nat.prime",
"pow_one"
] | If `irreducible (cyclotomic (p ^ n) R)` then `irreducible (cyclotomic p R).` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_mul_prime_eq_pow_of_not_dvd (R : Type*) {p n : ℕ} [hp : fact (nat.prime p)]
[ring R] [char_p R p] (hn : ¬p ∣ n) : cyclotomic (n * p) R = (cyclotomic n R) ^ (p - 1) | begin
letI : algebra (zmod p) R := zmod.algebra _ _,
suffices : cyclotomic (n * p) (zmod p) = (cyclotomic n (zmod p)) ^ (p - 1),
{ rw [← map_cyclotomic _ (algebra_map (zmod p) R), ← map_cyclotomic _ (algebra_map (zmod p) R),
this, polynomial.map_pow] },
apply mul_right_injective₀ (cyclotomic_ne_zero n $ z... | lemma | polynomial.cyclotomic_mul_prime_eq_pow_of_not_dvd | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/expand.lean | [
"ring_theory.polynomial.cyclotomic.roots"
] | [
"algebra",
"algebra_map",
"char_p",
"fact",
"mul_comm",
"mul_right_injective₀",
"nat.prime",
"polynomial.map_mul",
"polynomial.map_pow",
"ring",
"tsub_add_cancel_of_le",
"zmod",
"zmod.algebra",
"zmod.expand_card"
] | If `R` is of characteristic `p` and `¬p ∣ n`, then
`cyclotomic (n * p) R = (cyclotomic n R) ^ (p - 1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_mul_prime_dvd_eq_pow (R : Type*) {p n : ℕ} [hp : fact (nat.prime p)] [ring R]
[char_p R p] (hn : p ∣ n) : cyclotomic (n * p) R = (cyclotomic n R) ^ p | begin
letI : algebra (zmod p) R := zmod.algebra _ _,
suffices : cyclotomic (n * p) (zmod p) = (cyclotomic n (zmod p)) ^ p,
{ rw [← map_cyclotomic _ (algebra_map (zmod p) R), ← map_cyclotomic _ (algebra_map (zmod p) R),
this, polynomial.map_pow] },
rw [← zmod.expand_card, ← map_cyclotomic_int n, ← map_expa... | lemma | polynomial.cyclotomic_mul_prime_dvd_eq_pow | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/expand.lean | [
"ring_theory.polynomial.cyclotomic.roots"
] | [
"algebra",
"algebra_map",
"char_p",
"fact",
"mul_comm",
"nat.prime",
"polynomial.map_pow",
"ring",
"zmod",
"zmod.algebra",
"zmod.expand_card"
] | If `R` is of characteristic `p` and `p ∣ n`, then
`cyclotomic (n * p) R = (cyclotomic n R) ^ p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_mul_prime_pow_eq (R : Type*) {p m : ℕ} [fact (nat.prime p)]
[ring R] [char_p R p] (hm : ¬p ∣ m) :
∀ {k}, 0 < k → cyclotomic (p ^ k * m) R = (cyclotomic m R) ^ (p ^ k - p ^ (k - 1)) | | 1 _ := by rw [pow_one, nat.sub_self, pow_zero, mul_comm,
cyclotomic_mul_prime_eq_pow_of_not_dvd R hm]
| (a + 2) _ :=
begin
have hdiv : p ∣ p ^ a.succ * m := ⟨p ^ a * m, by rw [← mul_assoc, pow_succ]⟩,
rw [pow_succ, mul_assoc, mul_comm, cyclotomic_mul_prime_dvd_eq_pow R hdiv,
cyclotomic_mul_prime_pow_eq a.... | lemma | polynomial.cyclotomic_mul_prime_pow_eq | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/expand.lean | [
"ring_theory.polynomial.cyclotomic.roots"
] | [
"char_p",
"fact",
"mul_assoc",
"mul_comm",
"nat.prime",
"pow_mul",
"pow_one",
"pow_succ",
"pow_succ'",
"pow_zero",
"ring",
"tsub_zero"
] | If `R` is of characteristic `p` and `¬p ∣ m`, then
`cyclotomic (p ^ k * m) R = (cyclotomic m R) ^ (p ^ k - p ^ (k - 1))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_root_cyclotomic_prime_pow_mul_iff_of_char_p {m k p : ℕ} {R : Type*} [comm_ring R]
[is_domain R] [hp : fact (nat.prime p)] [hchar : char_p R p] {μ : R} [ne_zero (m : R)] :
(polynomial.cyclotomic (p ^ k * m) R).is_root μ ↔ is_primitive_root μ m | begin
letI : algebra (zmod p) R := zmod.algebra _ _,
rcases k.eq_zero_or_pos with rfl | hk,
{ rw [pow_zero, one_mul, is_root_cyclotomic_iff] },
refine ⟨λ h, _, λ h, _⟩,
{ rw [is_root.def, cyclotomic_mul_prime_pow_eq R (ne_zero.not_char_dvd R p m) hk, eval_pow] at h,
replace h := pow_eq_zero h,
rwa [← ... | lemma | polynomial.is_root_cyclotomic_prime_pow_mul_iff_of_char_p | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/expand.lean | [
"ring_theory.polynomial.cyclotomic.roots"
] | [
"algebra",
"char_p",
"comm_ring",
"fact",
"is_domain",
"is_primitive_root",
"nat.prime",
"ne_zero",
"ne_zero.not_char_dvd",
"one_mul",
"polynomial.cyclotomic",
"pow_eq_zero",
"pow_strict_mono_right",
"pow_zero",
"tsub_pos_iff_lt",
"zero_pow",
"zmod",
"zmod.algebra"
] | If `R` is of characteristic `p` and `¬p ∣ m`, then `ζ` is a root of `cyclotomic (p ^ k * m) R`
if and only if it is a primitive `m`-th root of unity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_root_of_unity_of_root_cyclotomic {ζ : R} {i : ℕ}
(hi : i ∈ n.divisors) (h : (cyclotomic i R).is_root ζ) : ζ ^ n = 1 | begin
rcases n.eq_zero_or_pos with rfl | hn,
{ exact pow_zero _ },
have := congr_arg (eval ζ) (prod_cyclotomic_eq_X_pow_sub_one hn R).symm,
rw [eval_sub, eval_pow, eval_X, eval_one] at this,
convert eq_add_of_sub_eq' this,
convert (add_zero _).symm,
apply eval_eq_zero_of_dvd_of_eval_eq_zero _ h,
exact f... | lemma | polynomial.is_root_of_unity_of_root_cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"finset.dvd_prod_of_mem",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_root_of_unity_iff (h : 0 < n) (R : Type*) [comm_ring R] [is_domain R]
{ζ : R} : ζ ^ n = 1 ↔ ∃ i ∈ n.divisors, (cyclotomic i R).is_root ζ | by rw [←mem_nth_roots h, nth_roots, mem_roots $ X_pow_sub_C_ne_zero h _,
C_1, ←prod_cyclotomic_eq_X_pow_sub_one h, is_root_prod]; apply_instance | lemma | is_root_of_unity_iff | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"comm_ring",
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_primitive_root.is_root_cyclotomic (hpos : 0 < n) {μ : R}
(h : is_primitive_root μ n) : is_root (cyclotomic n R) μ | begin
rw [← mem_roots (cyclotomic_ne_zero n R),
cyclotomic_eq_prod_X_sub_primitive_roots h, roots_prod_X_sub_C, ← finset.mem_def],
rwa [← mem_primitive_roots hpos] at h,
end | lemma | is_primitive_root.is_root_cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"finset.mem_def",
"is_primitive_root",
"mem_primitive_roots"
] | Any `n`-th primitive root of unity is a root of `cyclotomic n R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_root_cyclotomic_iff' {n : ℕ} {K : Type*} [field K] {μ : K} [ne_zero (n : K)] :
is_root (cyclotomic n K) μ ↔ is_primitive_root μ n | begin
-- in this proof, `o` stands for `order_of μ`
have hnpos : 0 < n := (ne_zero.of_ne_zero_coe K).out.bot_lt,
refine ⟨λ hμ, _, is_primitive_root.is_root_cyclotomic hnpos⟩,
have hμn : μ ^ n = 1,
{ rw is_root_of_unity_iff hnpos _,
exact ⟨n, n.mem_divisors_self hnpos.ne', hμ⟩,
all_goals { apply_instan... | lemma | polynomial.is_root_cyclotomic_iff' | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"by_contra",
"field",
"finset.insert_subset",
"finset.prod_pair",
"is_of_fin_order_iff_pow_eq_one",
"is_primitive_root",
"is_primitive_root.is_root_cyclotomic",
"is_primitive_root.not_iff",
"is_root_of_unity_iff",
"nat.dvd_of_mem_divisors",
"ne_zero",
"ne_zero.nat_cast_ne",
"ne_zero.of_ne_ze... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_root_cyclotomic_iff [ne_zero (n : R)] {μ : R} :
is_root (cyclotomic n R) μ ↔ is_primitive_root μ n | begin
have hf : function.injective _ := is_fraction_ring.injective R (fraction_ring R),
haveI : ne_zero (n : fraction_ring R) := ne_zero.nat_of_injective hf,
rw [←is_root_map_iff hf, ←is_primitive_root.map_iff_of_injective hf, map_cyclotomic,
←is_root_cyclotomic_iff']
end | lemma | polynomial.is_root_cyclotomic_iff | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"fraction_ring",
"is_fraction_ring.injective",
"is_primitive_root",
"ne_zero",
"ne_zero.nat_of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_cyclotomic_nodup [ne_zero (n : R)] : (cyclotomic n R).roots.nodup | begin
obtain h | ⟨ζ, hζ⟩ := (cyclotomic n R).roots.empty_or_exists_mem,
{ exact h.symm ▸ multiset.nodup_zero },
rw [mem_roots $ cyclotomic_ne_zero n R, is_root_cyclotomic_iff] at hζ,
refine multiset.nodup_of_le (roots.le_of_dvd (X_pow_sub_C_ne_zero
(ne_zero.pos_of_ne_zero_coe R) 1) $ cyclotomic.dvd_X_pow_su... | lemma | polynomial.roots_cyclotomic_nodup | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"multiset.nodup_of_le",
"multiset.nodup_zero",
"ne_zero",
"ne_zero.pos_of_ne_zero_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic.roots_to_finset_eq_primitive_roots [ne_zero (n : R)] :
(⟨(cyclotomic n R).roots, roots_cyclotomic_nodup⟩ : finset _) = primitive_roots n R | by { ext, simp [cyclotomic_ne_zero n R, is_root_cyclotomic_iff,
mem_primitive_roots, ne_zero.pos_of_ne_zero_coe R] } | lemma | polynomial.cyclotomic.roots_to_finset_eq_primitive_roots | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"finset",
"mem_primitive_roots",
"ne_zero",
"ne_zero.pos_of_ne_zero_coe",
"primitive_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic.roots_eq_primitive_roots_val [ne_zero (n : R)] :
(cyclotomic n R).roots = (primitive_roots n R).val | by rw ←cyclotomic.roots_to_finset_eq_primitive_roots | lemma | polynomial.cyclotomic.roots_eq_primitive_roots_val | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"ne_zero",
"primitive_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_root_cyclotomic_iff_char_zero {n : ℕ} {R : Type*} [comm_ring R] [is_domain R]
[char_zero R] {μ : R} (hn : 0 < n) :
(polynomial.cyclotomic n R).is_root μ ↔ is_primitive_root μ n | by { letI := ne_zero.of_gt hn, exact is_root_cyclotomic_iff } | lemma | polynomial.is_root_cyclotomic_iff_char_zero | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"char_zero",
"comm_ring",
"is_domain",
"is_primitive_root",
"ne_zero.of_gt",
"polynomial.cyclotomic"
] | If `R` is of characteristic zero, then `ζ` is a root of `cyclotomic n R` if and only if it is a
primitive `n`-th root of unity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_injective [char_zero R] :
function.injective (λ n, cyclotomic n R) | begin
intros n m hnm,
simp only at hnm,
rcases eq_or_ne n 0 with rfl | hzero,
{ rw [cyclotomic_zero] at hnm,
replace hnm := congr_arg nat_degree hnm,
rw [nat_degree_one, nat_degree_cyclotomic] at hnm,
by_contra,
exact (nat.totient_pos (zero_lt_iff.2 (ne.symm h))).ne hnm },
{ haveI := ne_zero.m... | lemma | polynomial.cyclotomic_injective | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"by_contra",
"char_zero",
"complex.is_primitive_root_exp",
"eq_or_ne",
"int.cast_injective",
"int.cast_ring_hom",
"is_primitive_root.eq_order_of",
"nat.totient_pos",
"ne_zero"
] | Over a ring `R` of characteristic zero, `λ n, cyclotomic n R` is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_primitive_root.minpoly_dvd_cyclotomic {n : ℕ} {K : Type*} [field K] {μ : K}
(h : is_primitive_root μ n) (hpos : 0 < n) [char_zero K] :
minpoly ℤ μ ∣ cyclotomic n ℤ | begin
apply minpoly.is_integrally_closed_dvd (h.is_integral hpos),
simpa [aeval_def, eval₂_eq_eval_map, is_root.def] using h.is_root_cyclotomic hpos
end | lemma | is_primitive_root.minpoly_dvd_cyclotomic | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"char_zero",
"field",
"is_primitive_root",
"minpoly",
"minpoly.is_integrally_closed_dvd"
] | The minimal polynomial of a primitive `n`-th root of unity `μ` divides `cyclotomic n ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_primitive_root.minpoly_eq_cyclotomic_of_irreducible {K : Type*} [field K]
{R : Type*} [comm_ring R] [is_domain R] {μ : R} {n : ℕ} [algebra K R] (hμ : is_primitive_root μ n)
(h : irreducible $ cyclotomic n K) [ne_zero (n : K)] : cyclotomic n K = minpoly K μ | begin
haveI := ne_zero.of_no_zero_smul_divisors K R n,
refine minpoly.eq_of_irreducible_of_monic h _ (cyclotomic.monic n K),
rwa [aeval_def, eval₂_eq_eval_map, map_cyclotomic, ←is_root.def, is_root_cyclotomic_iff]
end | lemma | is_primitive_root.minpoly_eq_cyclotomic_of_irreducible | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"algebra",
"comm_ring",
"field",
"irreducible",
"is_domain",
"is_primitive_root",
"minpoly",
"minpoly.eq_of_irreducible_of_monic",
"ne_zero",
"ne_zero.of_no_zero_smul_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cyclotomic_eq_minpoly {n : ℕ} {K : Type*} [field K] {μ : K}
(h : is_primitive_root μ n) (hpos : 0 < n) [char_zero K] :
cyclotomic n ℤ = minpoly ℤ μ | begin
refine eq_of_monic_of_dvd_of_nat_degree_le (minpoly.monic (is_integral h hpos))
(cyclotomic.monic n ℤ) (h.minpoly_dvd_cyclotomic hpos) _,
simpa [nat_degree_cyclotomic n ℤ] using totient_le_degree_minpoly h
end | lemma | polynomial.cyclotomic_eq_minpoly | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"char_zero",
"field",
"is_integral",
"is_primitive_root",
"minpoly",
"minpoly.monic"
] | `cyclotomic n ℤ` is the minimal polynomial of a primitive `n`-th root of unity `μ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic_eq_minpoly_rat {n : ℕ} {K : Type*} [field K] {μ : K}
(h : is_primitive_root μ n) (hpos : 0 < n) [char_zero K] :
cyclotomic n ℚ = minpoly ℚ μ | begin
rw [← map_cyclotomic_int, cyclotomic_eq_minpoly h hpos],
exact (minpoly.is_integrally_closed_eq_field_fractions' _ (is_integral h hpos)).symm
end | lemma | polynomial.cyclotomic_eq_minpoly_rat | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"char_zero",
"field",
"is_integral",
"is_primitive_root",
"minpoly",
"minpoly.is_integrally_closed_eq_field_fractions'"
] | `cyclotomic n ℚ` is the minimal polynomial of a primitive `n`-th root of unity `μ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic.irreducible {n : ℕ} (hpos : 0 < n) : irreducible (cyclotomic n ℤ) | begin
rw [cyclotomic_eq_minpoly (is_primitive_root_exp n hpos.ne') hpos],
apply minpoly.irreducible,
exact (is_primitive_root_exp n hpos.ne').is_integral hpos,
end | lemma | polynomial.cyclotomic.irreducible | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"irreducible",
"is_integral",
"minpoly.irreducible"
] | `cyclotomic n ℤ` is irreducible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic.irreducible_rat {n : ℕ} (hpos : 0 < n) : irreducible (cyclotomic n ℚ) | begin
rw [← map_cyclotomic_int],
exact (is_primitive.irreducible_iff_irreducible_map_fraction_map (cyclotomic.is_primitive n ℤ)).1
(cyclotomic.irreducible hpos),
end | lemma | polynomial.cyclotomic.irreducible_rat | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"irreducible"
] | `cyclotomic n ℚ` is irreducible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cyclotomic.is_coprime_rat {n m : ℕ} (h : n ≠ m) :
is_coprime (cyclotomic n ℚ) (cyclotomic m ℚ) | begin
rcases n.eq_zero_or_pos with rfl | hnzero,
{ exact is_coprime_one_left },
rcases m.eq_zero_or_pos with rfl | hmzero,
{ exact is_coprime_one_right },
rw (irreducible.coprime_iff_not_dvd $ cyclotomic.irreducible_rat $ hnzero),
exact (λ hdiv, h $ cyclotomic_injective $ eq_of_monic_of_associated (cyclotom... | lemma | polynomial.cyclotomic.is_coprime_rat | ring_theory.polynomial.cyclotomic | src/ring_theory/polynomial/cyclotomic/roots.lean | [
"ring_theory.polynomial.cyclotomic.basic",
"ring_theory.roots_of_unity.minpoly"
] | [
"irreducible.associated_of_dvd",
"irreducible.coprime_iff_not_dvd",
"is_coprime",
"is_coprime_one_left",
"is_coprime_one_right"
] | If `n ≠ m`, then `(cyclotomic n ℚ)` and `(cyclotomic m ℚ)` are coprime. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_weakly_eisenstein_at [comm_semiring R] (f : R[X]) (𝓟 : ideal R) :
Prop | (mem : ∀ {n}, n < f.nat_degree → f.coeff n ∈ 𝓟) | structure | polynomial.is_weakly_eisenstein_at | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"comm_semiring",
"ideal"
] | Given an ideal `𝓟` of a commutative semiring `R`, we say that a polynomial `f : R[X]`
is *weakly Eisenstein at `𝓟`* if `∀ n, n < f.nat_degree → f.coeff n ∈ 𝓟`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_eisenstein_at [comm_semiring R] (f : R[X]) (𝓟 : ideal R) : Prop | (leading : f.leading_coeff ∉ 𝓟)
(mem : ∀ {n}, n < f.nat_degree → f.coeff n ∈ 𝓟)
(not_mem : f.coeff 0 ∉ 𝓟 ^ 2) | structure | polynomial.is_eisenstein_at | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"comm_semiring",
"ideal"
] | Given an ideal `𝓟` of a commutative semiring `R`, we say that a polynomial `f : R[X]`
is *Eisenstein at `𝓟`* if `f.leading_coeff ∉ 𝓟`, `∀ n, n < f.nat_degree → f.coeff n ∈ 𝓟` and
`f.coeff 0 ∉ 𝓟 ^ 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {A : Type v} [comm_ring A] (φ : R →+* A) : (f.map φ).is_weakly_eisenstein_at (𝓟.map φ) | begin
refine (is_weakly_eisenstein_at_iff _ _).2 (λ n hn, _),
rw [coeff_map],
exact mem_map_of_mem _ (hf.mem (lt_of_lt_of_le hn (nat_degree_map_le _ _)))
end | lemma | polynomial.is_weakly_eisenstein_at.map | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_adjoin_mul_eq_pow_nat_degree {x : S} (hx : aeval x f = 0)
(hmo : f.monic) (hf : f.is_weakly_eisenstein_at P) : ∃ y ∈ adjoin R ({x} : set S),
(algebra_map R S) p * y = x ^ (f.map (algebra_map R S)).nat_degree | begin
rw [aeval_def, polynomial.eval₂_eq_eval_map, eval_eq_sum_range, range_add_one,
sum_insert not_mem_range_self, sum_range, (hmo.map
(algebra_map R S)).coeff_nat_degree, one_mul] at hx,
replace hx := eq_neg_of_add_eq_zero_left hx,
have : ∀ n < f.nat_degree, p ∣ f.coeff n,
{ intros n hn,
refine me... | lemma | polynomial.is_weakly_eisenstein_at.exists_mem_adjoin_mul_eq_pow_nat_degree | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"algebra_map",
"fin.coe_eq_val",
"mul_assoc",
"mul_comm",
"neg_eq_neg_one_mul",
"one_mul",
"polynomial.eval₂_eq_eval_map",
"ring_hom.map_mul",
"set.mem_singleton",
"subalgebra.algebra_map_mem",
"subalgebra.mul_mem",
"subalgebra.neg_mem",
"subalgebra.one_mem",
"subalgebra.pow_mem",
"subal... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_adjoin_mul_eq_pow_nat_degree_le {x : S} (hx : aeval x f = 0)
(hmo : f.monic) (hf : f.is_weakly_eisenstein_at P) :
∀ i, (f.map (algebra_map R S)).nat_degree ≤ i →
∃ y ∈ adjoin R ({x} : set S), (algebra_map R S) p * y = x ^ i | begin
intros i hi,
obtain ⟨k, hk⟩ := exists_add_of_le hi,
rw [hk, pow_add],
obtain ⟨y, hy, H⟩ := exists_mem_adjoin_mul_eq_pow_nat_degree hx hmo hf,
refine ⟨y * x ^ k, _, _⟩,
{ exact subalgebra.mul_mem _ hy (subalgebra.pow_mem _ (subset_adjoin (set.mem_singleton x)) _) },
{ rw [← mul_assoc _ y, H] }
end | lemma | polynomial.is_weakly_eisenstein_at.exists_mem_adjoin_mul_eq_pow_nat_degree_le | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"algebra_map",
"mul_assoc",
"pow_add",
"set.mem_singleton",
"subalgebra.mul_mem",
"subalgebra.pow_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_nat_degree_le_of_root_of_monic_mem {x : R} (hroot : is_root f x) (hmo : f.monic) :
∀ i, f.nat_degree ≤ i → x ^ i ∈ 𝓟 | begin
intros i hi,
obtain ⟨k, hk⟩ := exists_add_of_le hi,
rw [hk, pow_add],
suffices : x ^ f.nat_degree ∈ 𝓟,
{ exact mul_mem_right (x ^ k) 𝓟 this },
rw [is_root.def, eval_eq_sum_range, finset.range_add_one, finset.sum_insert
finset.not_mem_range_self, finset.sum_range, hmo.coeff_nat_degree, one_mul] a... | lemma | polynomial.is_weakly_eisenstein_at.pow_nat_degree_le_of_root_of_monic_mem | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"fin.is_lt",
"finset.not_mem_range_self",
"finset.range_add_one",
"one_mul",
"pow_add",
"submodule.sum_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_nat_degree_le_of_aeval_zero_of_monic_mem_map {x : S} (hx : aeval x f = 0)
(hmo : f.monic) :
∀ i, (f.map (algebra_map R S)).nat_degree ≤ i → x ^ i ∈ 𝓟.map (algebra_map R S) | begin
suffices : x ^ (f.map (algebra_map R S)).nat_degree ∈ 𝓟.map (algebra_map R S),
{ intros i hi,
obtain ⟨k, hk⟩ := exists_add_of_le hi,
rw [hk, pow_add],
refine mul_mem_right _ _ this },
rw [aeval_def, eval₂_eq_eval_map, ← is_root.def] at hx,
refine pow_nat_degree_le_of_root_of_monic_mem (hf.map... | lemma | polynomial.is_weakly_eisenstein_at.pow_nat_degree_le_of_aeval_zero_of_monic_mem_map | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"algebra_map",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
scale_roots.is_weakly_eisenstein_at (p : R[X]) {x : R}
{P : ideal R} (hP : x ∈ P) : (scale_roots p x).is_weakly_eisenstein_at P | begin
refine ⟨λ i hi, _⟩,
rw coeff_scale_roots,
rw [nat_degree_scale_roots, ← tsub_pos_iff_lt] at hi,
exact ideal.mul_mem_left _ _ (ideal.pow_mem_of_mem P hP _ hi)
end | lemma | polynomial.scale_roots.is_weakly_eisenstein_at | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"ideal",
"ideal.mul_mem_left",
"ideal.pow_mem_of_mem",
"tsub_pos_iff_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_pow_nat_degree_of_eval₂_eq_zero {f : R →+* A}
(hf : function.injective f) {p : R[X]} (hp : p.monic) (x y : R) (z : A)
(h : p.eval₂ f z = 0) (hz : f x * z = f y) : x ∣ y ^ p.nat_degree | begin
rw [← nat_degree_scale_roots p x, ← ideal.mem_span_singleton],
refine (scale_roots.is_weakly_eisenstein_at _ (ideal.mem_span_singleton.mpr $ dvd_refl x))
.pow_nat_degree_le_of_root_of_monic_mem _ ((monic_scale_roots_iff x).mpr hp) _ le_rfl,
rw injective_iff_map_eq_zero' at hf,
have := scale_roots_eval... | lemma | polynomial.dvd_pow_nat_degree_of_eval₂_eq_zero | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"dvd_refl",
"ideal.mem_span_singleton",
"le_rfl",
"polynomial.eval₂_at_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_pow_nat_degree_of_aeval_eq_zero [algebra R A] [nontrivial A]
[no_zero_smul_divisors R A] {p : R[X]} (hp : p.monic) (x y : R) (z : A)
(h : polynomial.aeval z p = 0) (hz : z * algebra_map R A x = algebra_map R A y) :
x ∣ y ^ p.nat_degree | dvd_pow_nat_degree_of_eval₂_eq_zero (no_zero_smul_divisors.algebra_map_injective R A)
hp x y z h ((mul_comm _ _).trans hz) | lemma | polynomial.dvd_pow_nat_degree_of_aeval_eq_zero | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"algebra",
"algebra_map",
"mul_comm",
"no_zero_smul_divisors",
"no_zero_smul_divisors.algebra_map_injective",
"nontrivial",
"polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.polynomial.monic.leading_coeff_not_mem (hf : f.monic) (h : 𝓟 ≠ ⊤) :
¬f.leading_coeff ∈ 𝓟 | hf.leading_coeff.symm ▸ (ideal.ne_top_iff_one _).1 h | lemma | polynomial.monic.leading_coeff_not_mem | ring_theory.polynomial.eisenstein | src/ring_theory/polynomial/eisenstein/basic.lean | [
"ring_theory.eisenstein_criterion",
"ring_theory.polynomial.scale_roots"
] | [
"ideal.ne_top_iff_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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