text stringlengths 2 132k | source dict |
|---|---|
embeddings into CIF instead of RIF. EXAMPLES: Sage sage: x = polygen(QQ, 'x') sage: F. = NumberField(x^3 - 100*x + 1); F.places() [Ring morphism: From: Number Field in alpha with defining polynomial x^3 - 100*x + 1 To: Real Field with 106 bits of precision Defn: alpha |--> -10.00499625499181184573367219280, Ring morphi... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
sage: F. = NumberField(x^4 - 7) ; F.real_places() [Ring morphism: From: Number Field in alpha with defining polynomial x^4 - 7 To: Real Field with 106 bits of precision Defn: alpha |--> -1.626576561697785743211232345494, Ring morphism: From: Number Field in alpha with defining polynomial x^4 - 7 To: Real Field with 106... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
self names β 2-tuple of names of generator for output field πΎ and the subfield π ( πΌ ) structure β an instance of structure.NumberFieldStructure or None (default: None), if None, then the resulting fieldβs structure() will return isomorphisms from and to this field. Otherwise, the field will be equipped with structu... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
False sage: L1.base_field() is K0 True sage: L2.base_field() is K0 True Python Here we see that with the different embeddings, the relative norms are different: Sage sage: a0 = K0.gen() sage: L1_into_K, K_into_L1 = L1.structure() sage: L2_into_K, K_into_L2 = L2.structure() sage: len(K.factor(41)) 4 sage: w1 = -a^2 + a ... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
Defn: c1 |--> c] sage: len(L.subfields(2)) 0 sage: len(L.subfields(1)) 1 Python sage.rings.number_field.number_field.NumberField_absolute_v1(poly, name, latex_name, canonical_embedding=None) [source] Used for unpickling old pickles. EXAMPLES: Sage sage: from sage.rings.number_field.number_field import NumberField_absol... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
in MPFR) means that only the prec bits of the number after the decimal point are valid. Sage sage: K = CyclotomicField(3) sage: phi = K.complex_embedding(10) sage: phi(K.0) -0.50 + 0.87*I sage: phi(K.0^3) 1.0 sage: phi(K.0^3 - 1) 0.00 sage: phi(K.0^3 + 7) 8.0 Python complex_embeddings(prec=53) [source] Return all embed... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
list v. Uses the formula for the discriminant of a prime power cyclotomic field and Hilbert Theorem 88 on the discriminant of composita. INPUT: v β (optional) list of elements of this number field OUTPUT: integer if v is omitted, and Rational otherwise EXAMPLES: Sage sage: CyclotomicField(20).discriminant() 4000000 sag... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
field into the approximate real field with precision prec. Mostly, of course, there are no such embeddings. EXAMPLES: Sage sage: len(CyclotomicField(4).real_embeddings()) 0 sage: CyclotomicField(2).real_embeddings() [Ring morphism: From: Cyclotomic Field of order 2 and degree 1 To: Real Field with 53 bits of precision ... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
-a, -a^2, -a^3] sage: K.zeta(2, all=True) [-1] sage: K. = CyclotomicField(10) sage: K.zeta(20, all=True) Traceback (most recent call last): ... ValueError: 20 does not divide order of generator (10) Python Sage sage: K. = CyclotomicField(5) sage: K.zeta(4) Traceback (most recent call last): ... ValueError: 4 does not d... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
we include the prime ( 2 , π + 1 ) , the S-class group becomes trivial: Sage sage: K.S_class_group([K.ideal(2, a + 1)]) S-class group of order 1 of Number Field in a with defining polynomial x^2 + 5 with a = 2.236067977499790?*I Python S_unit_group(proof=None, S=None) [source] Return the π -unit group (including tors... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
sage: U = K.unit_group(proof=False) sage: U.is_isomorphic(K.S_unit_group(proof=False)) True Python The value of π may be specified as a list of prime ideals, or an ideal, or an element of the field: Sage sage: K. = NumberField(x^3 + 3) sage: U = K.S_unit_group(proof=False, S=K.ideal(6).prime_factors()); U S-unit group... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
π₯ + π¦ = 1 over self. INPUT: S β list of finite primes in this number field prec β precision used for computations in real, complex, and π -adic fields (default: 106) include_exponents β whether to include the exponent vectors in the returned value (default: True) include_bound β whether to return the final computed ... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
sage: K.S_unit_solutions(S) # random, due to ordering [(xi + 2, -xi - 1), (1/3*xi + 2/3, -1/3*xi + 1/3), (-xi, xi + 1), (-xi + 1, xi)] Python You can get the exponent vectors: Sage sage: # needs sage.rings.padics sage: K.S_unit_solutions(S, include_exponents=True) # random, due to ordering [((2, 1), (4, 0), xi + 2, -xi... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
NumberField(x^3 + x^2 + 997*x + 1, 'a').absolute_degree() 3 sage: NumberField(x + 1, 'a').absolute_degree() 1 sage: NumberField(x^997 + 17*x + 3, 'a', check=False).absolute_degree() 997 Python absolute_field(names) [source] Return self as an absolute number field. INPUT: names β string; name of generator of the absolut... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
i0 |--> 1/2*i + 3 sage: to_K Ring morphism: From: Number Field in i with defining polynomial x^2 + 1 To: Number Field in i0 with defining polynomial x^2 - 6*x + 37/4 with i0 = 1/2*i + 3 Defn: i |--> 2*i0 - 6 Python We can also do Sage sage: K., from_K, to_K = L.change_generator(i/2 + 3); K Number Field in c with defini... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
ideal class (2, 1/2*a - 1/2),) Python There can be multiple generators: Sage sage: k. = NumberField(x^2 + 20072) sage: G = k.class_group(); G Class group of order 76 with structure C38 x C2 of Number Field in a with defining polynomial x^2 + 20072 sage: G.0 # random Fractional ideal class (41, a + 10) sage: G.0^38 Triv... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
71, 73, 89] Python completion(p, prec, extras={}) [source] Return the completion of self at π to the specified precision. Only implemented at archimedean places, and then only if an embedding has been fixed. EXAMPLES: Sage sage: K. = QuadraticField(2) sage: K.completion(infinity, 100) Real Field with 100 bits of preci... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
with precision prec. This always embeds into an MPFR based complex field. If you want embeddings into the 53-bit double precision, which is faster, use self.embeddings(CDF). EXAMPLES: Sage sage: x = polygen(QQ, 'x') sage: k. = NumberField(x^5 + x + 17) sage: v = k.complex_embeddings() sage: ls = [phi(k.0^2) for phi in ... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
if self and other both have embeddings into the same ambient field, or into fields which are contained in a common field, only the compositum respecting both embeddings is returned. In all other cases, all possible composite number fields are returned. EXAMPLES: Sage sage: x = polygen(QQ, 'x') sage: K. = NumberField(x^... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
16*x^2 + 4) sage: Q1.composite_fields(Q2, 'c') [Number Field in c with defining polynomial x^8 + 64*x^6 + 904*x^4 + 3840*x^2 + 3600] sage: F, Q1_into_F, Q2_into_F, k = Q1.composite_fields(Q2, 'c', ....: both_maps=True) sage: Q1_into_F Ring morphism: From: Number Field in a with defining polynomial x^4 + 10*x^2 + 1 To: ... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
2000*x^2 + 50, 'a').conductor() 160 Python ALGORITHM: For odd primes, it is easy to compute from the ramification index because the π -Sylow subgroup is cyclic. For π = 2 , there are two choices for a given ramification index. They can be distinguished by the parity of the exponent in the discriminant of a 2-adic com... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
3^288 * 7^18 * 11^17 * 31^18 * 157^18 * 2153^18 * 13907^18 * ... sage: K.decomposition_type(2) [(2, 4, 3), (2, 12, 2), (2, 36, 1)] sage: K.decomposition_type(3) [(9, 3, 2), (9, 10, 1)] sage: K.decomposition_type(7) [(1, 18, 1), (1, 90, 1), (2, 1, 6), (2, 3, 4)] Python It also works for relative extensions: Sage sage: K... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
'x') sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').degree() 3 sage: NumberField(x + 1, 'a').degree() 1 sage: NumberField(x^997 + 17*x + 3, 'a', check=False).degree() 997 Python different() [source] Compute the different fractional ideal of this number field. The codifferent is the fractional ideal of all π₯ in πΎ such ... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
zeta5, Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta5^2, Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta5^3, Dirichlet character modulo 11 of conductor 11 mapping 2 |--> -zeta5^3 - zeta5^2 - zeta5 - 1] sage: X^2 # optional - gap_package_polycyclic Dirichlet character modulo 11 of c... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
b with defining polynomial t^2 + a over its base field Python We create another extension: Sage sage: k. = NumberField(x^2 + 1); k Number Field in a with defining polynomial x^2 + 1 sage: y = polygen(QQ,'y') sage: m. = k.extension(y^2 + 2); m Number Field in b with defining polynomial y^2 + 2 over its base field Python... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
or ideals of the number field: Sage sage: K. = NumberField(x^2 + 1) sage: K.factor(1/3) (Fractional ideal (3))^-1 sage: K.factor(1+a) Fractional ideal (a - 1) sage: K.factor(1+a/5) (Fractional ideal (a - 1)) * (Fractional ideal (2*a - 1))^-1 * (Fractional ideal (-2*a - 1))^-1 * (Fractional ideal (3*a + 2)) Python An ex... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
element (zero ideal is not a fractional ideal) Python galois_group(type=None, algorithm='pari', names=None, gc_numbering=None) [source] Return the Galois group of the Galois closure of this number field. INPUT: type β deprecated; the different versions of Galois groups have been merged in Issue #28782 algorithm β 'pari... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
group number 2 of degree 3 of the Number Field in a with defining polynomial x^3 + 2*x + 1 Python EXPLICIT GALOIS GROUP: We compute the Galois group as an explicit group of automorphisms of the Galois closure of a field. Sage sage: # needs sage.groups sage: K. = NumberField(x^3 - 2) sage: L. = K.galois_closure(); L Num... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
= NumberField(x^14 + 2); k Number Field in theta with defining polynomial x^14 + 2 sage: k.gen() theta sage: k.gen(1) Traceback (most recent call last): ... IndexError: Only one generator. Python gen_embedding() [source] If an embedding has been specified, return the image of the generator under that embedding. Otherwi... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
-63/2*sqrt2 - 20 sage: all( ....: (r - a).valuation(P) >= k ....: for I, a in zip(ideals, residues) ....: for P, k in I.factor() ....: ) True Python ideals_of_bdd_norm(bound) [source] Return all integral ideals of bounded norm. INPUT: bound β positive integer OUTPUT: a dict of all integral ideals πΌ such that Norm( πΌ ... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
a cubic field in which 2 is an βessential discriminant divisorβ, so the ring of integers is not generated by a single element. Sage sage: K. = NumberField(x^3 + x^2 - 2*x + 8) sage: K.integral_basis() [1, 1/2*a^2 + 1/2*a, a^2] Python ALGORITHM: Uses the PARI library (via pari:_pari_integral_basis). is_CM() [source] Ret... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
sage: y = polygen(E_0) sage: E. = E_0.extension(y^2 - E_0.gen() / 2) sage: E.is_CM() True sage: E.is_CM_extension() False Python is_abelian() [source] Return True if this number field is an abelian Galois extension of π . EXAMPLES: Sage sage: # needs sage.groups sage: x = polygen(QQ, 'x') sage: NumberField(x^2 + 1, 'i... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
+ 4) sage: k.is_isomorphic(m) True sage: m. = NumberField(x^2 + 5) sage: k.is_isomorphic (m) False Python Sage sage: k = NumberField(x^3 + 2, 'a') sage: k.is_isomorphic(NumberField((x+1/3)^3 + 2, 'b')) True sage: k.is_isomorphic(NumberField(x^3 + 4, 'b')) True sage: k.is_isomorphic(NumberField(x^3 + 5, 'b')) False sage... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
this number field. INPUT: v β None, a prime, or a list of integer primes (default: None) if None, return the maximal order. if a prime π , return an order that is π -maximal. if a list, return an order that is maximal at each prime of these primes assume_maximal β True, False, None, or 'non-maximal-non-unique' (defau... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
with bigger discriminant: Sage sage: p = next_prime(10^97) sage: q = next_prime(10^99) sage: K. = NumberField(x^3 - p*q) sage: K.maximal_order(prime_range(10000), assume_maximal=None).basis() [1, a, a^2] Python An example in a relative number field: Sage sage: K. = NumberField([x^2 + 1, x^2 - 3]) sage: OK = K.maximal_o... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
210*x^6 + 126*x^5 - 126*x^4 - 56*x^3 + 28*x^2 + 7*x - 1 with a0 = 1.953241111420174?, Ring morphism: From: Number Field in a0 with defining polynomial x^14 + x^13 - 13*x^12 - 12*x^11 + 66*x^10 + 55*x^9 - 165*x^8 - 120*x^7 + 210*x^6 + 126*x^5 - 126*x^4 - 56*x^3 + 28*x^2 + 7*x - 1 with a0 = 1.953241111420174? To: Cycloto... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
1] Python An example of a relative extension where the base field is not the maximal totally real subfield. Sage sage: E_0. = NumberField(x^2 - 4*x + 16) sage: y = polygen(E_0) sage: E. = E_0.extension(y^2 - E_0.gen() / 2) sage: E.maximal_totally_real_subfield() [Number Field in z1 with defining polynomial x^2 - 2*x - ... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
F. = NumberField(x^22 + 3) sage: F.zeta_order() 6 sage: F. = NumberField(x^2 - 7) sage: F.zeta_order() 2 Python order() [source] Return the order of this number field (always +infinity). OUTPUT: always positive infinity EXAMPLES: Sage sage: x = polygen(QQ, 'x') sage: NumberField(x^2 + 19,'a').order() +Infinity Python p... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
sage: pari(k)[:4] [y^4 - 3*y + 7, [0, 2], 85621, 1] Python Sage sage: k. = NumberField(x^4 - 3/2*x + 5/3); k Number Field in a with defining polynomial x^4 - 3/2*x + 5/3 sage: k.pari_nf() [y^4 - 324*y + 2160, [0, 2], 48918708, 216, ..., [36, 36*y, y^3 + 6*y^2 - 252, -6*y^2], [1, 0, 0, 252; 0, 1, 0, 0; 0, 0, 0, 36; 0, 0... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
is not the same as simply converting the defining polynomial to PARI. EXAMPLES: Sage sage: y = polygen(QQ) sage: k. = NumberField(y^2 - 3/2*y + 5/3) sage: k.pari_polynomial() x^2 - x + 40 sage: k.polynomial().__pari__() x^2 - 3/2*x + 5/3 sage: k.pari_polynomial('a') a^2 - a + 40 Python Some examples with relative numbe... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
this number field. This is exactly the same as defining_polynomial(). EXAMPLES: Sage sage: x = polygen(QQ, 'x') sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial() x^2 + 2/3*x - 9/17 Python polynomial_ntl() [source] Return defining polynomial of this number field as a pair, an ntl polynomial and a denominator. Thi... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
EXAMPLES: Sage sage: x = polygen(QQ, 'x') sage: K. = NumberField(x^5 + 10*x + 1) sage: K.power_basis() [1, a, a^2, a^3, a^4] Python Sage sage: L. = K.extension(x^2 - 2) sage: L.power_basis() [1, b] sage: L.absolute_field('c').power_basis() [1, c, c^2, c^3, c^4, c^5, c^6, c^7, c^8, c^9] Python Sage sage: M = CyclotomicF... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
P5_2 = F.prime_above(5, degree=2) sage: P5_2 # random Fractional ideal (t^2 - 2*t - 1) sage: P5_2.residue_class_degree() 2 Python Relative number fields are ok: Sage sage: G = F.extension(x^2 - 11, 'b') sage: G.prime_above(7) Fractional ideal (b + 2) Python It doesnβt make sense to factor the ideal ( 0 ) : Sage sage: F... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
for P2 in P2s) True sage: all(P2.is_prime() for P2 in P2s) True sage: [ P2.norm() for P2 in P2s ] Python Sage sage: P3s = F.primes_above(3) sage: P3s # random [Fractional ideal (t + 1)] sage: all(3 in P3 for P3 in P3s) True sage: all(P3.is_prime() for P3 in P3s) True sage: [ P3.norm() for P3 in P3s ] Python The ideal... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
the same norm are sorted using the comparison function for ideals, which is based on the Hermite Normal Form. Note See also primes_of_bounded_norm_iter() for an iterator version of this, but note that the iterator sorts the primes in order of underlying rational prime, not by norm. EXAMPLES: Sage sage: K. = QuadraticFi... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
details. INPUT: num_integer_primes β (default: 10000) an integer. We try to find primes of absolute norm no greater than the num_integer_primes-th prime number. For example, if num_integer_primes is 2, the largest norm found will be 3, since the second prime is 3. max_iterations β (default: 100) an integer. We test max... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
Python primitive_element() [source] Return a primitive element for this field, i.e., an element that generates it over π . EXAMPLES: Sage sage: x = polygen(ZZ, 'x') sage: K. = NumberField(x^3 + 2) sage: K.primitive_element() a sage: K. = NumberField([x^2 - 2, x^2 - 3, x^2 - 5]) sage: K.primitive_element() a - b + c sa... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
element of self p β a prime ideal check β boolean (default: True); check if π is prime ALGORITHM: This is an implementation of Algorithm 3.1.3 from [Kir2016]. EXAMPLES: Sage sage: x = polygen(QQ, 'x') sage: K. = NumberField(x^2 + 2) sage: p = K.primes_above(2) sage: K.quadratic_defect(5, p) 4 sage: K.quadratic_defect(... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
precision real numbers. In functions that should output proven results, one could use self.embeddings(AA) instead. EXAMPLES: Sage sage: x = polygen(QQ, 'x') sage: K. = NumberField(x^3 + 2) sage: K.real_embeddings() [Ring morphism: From: Number Field in a with defining polynomial x^3 + 2 To: Real Field with 53 bits of p... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
t^5] sage: F.reduced_basis() [-1, -1/2*t^5 + 1/2*t^4 + 3*t^3 - 3/2*t^2 - 4*t - 1/2, t, 1/2*t^5 + 1/2*t^4 - 4*t^3 - 5/2*t^2 + 7*t + 1/2, 1/2*t^5 - 1/2*t^4 - 2*t^3 + 3/2*t^2 - 1/2, 1/2*t^5 - 1/2*t^4 - 3*t^3 + 5/2*t^2 + 4*t - 5/2] sage: CyclotomicField(12).reduced_basis() [1, zeta12^2, zeta12, zeta12^3] Python reduced_gra... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
-2] [ 0 0 14 6 -2 3] [ 2 1 6 16 -3 3] [ 0 0 -2 -3 16 6] [ 1 -2 3 3 6 19] sage: Matrix(6, [(x*y).trace() ....: for x in F.integral_basis() for y in F.integral_basis()]) [2550 133 259 664 1368 3421] [ 133 14 3 54 30 233] [ 259 3 54 30 233 217] [ 664 54 30 233 217 1078] [1368 30 233 217 1078 1371] [3421 233 217 1078 1371 ... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
Fractional ideal (i + 1) Python roots_of_unity() [source] Return all the roots of unity in this field, primitive or not. EXAMPLES: Sage sage: x = polygen(QQ, 'x') sage: K. = NumberField(x^2 + 1) sage: zs = K.roots_of_unity(); zs [b, -1, -b, 1] sage: [z**K.number_of_roots_of_unity() for z in zs] [1, 1, 1, 1] Python selm... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
πΉ π isomorphic to πΎ ( π , π ) and maps implementing the isomorphism between this space and πΎ ( π , π ) as a subgroup of πΎ β / ( πΎ β ) π . EXAMPLES: Sage sage: K. = QuadraticField(-5) sage: K.selmer_generators((), 2) [-1, 2] Python The previous example shows that the group generated by the output may be stric... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
group OUTPUT: An iterator yielding the distinct elements of πΎ ( π , π ) . See the docstring for NumberField_generic.selmer_generators() for more information. EXAMPLES: Sage sage: K. = QuadraticField(-5) sage: list(K.selmer_group_iterator((), 2)) [1, 2, -1, -2] sage: list(K.selmer_group_iterator((), 4)) [1, 4, -1, -4... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
The group πΎ ( π , π ) is the finite subgroup of πΎ β / ( πΎ β ) π consisting of elements whose valuation at all primes not in π is a multiple of π . It contains the subgroup of those π β πΎ β such that πΎ ( π π ) / πΎ is unramified at all primes of πΎ outside of π , but may contain it properly when not all pr... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
of πΎ unramified outside π , the Hilbert Class Field πΎ ( β 1 ) : Sage sage: K. = QuadraticField(-5) sage: KS2, gens, fromKS2, toKS2 = K.selmer_space([], 2) sage: KS2 Vector space of dimension 2 over Finite Field of size 2 sage: gens [2, -1] sage: x = polygen(ZZ, 'x') sage: for v in KS2: ....: if not v: ....: continue... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
sage: Ilist + Ilist Fractional ideal (2, a) Python some_elements() [source] Return a list of elements in the given number field. EXAMPLES: Sage sage: R. = QQ[] sage: K. = QQ.extension(t^2 - 2); K Number Field in a with defining polynomial t^2 - 2 sage: K.some_elements() [1, a, 2*a, 3*a - 4, 1/2, 1/3*a, 1/6*a, 0, 1/2*a,... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
-1.904160859134921? sage: NumberField(x^3 - x + 5, 'a', embedding=CDF.0).specified_complex_embedding() Generic morphism: From: Number Field in a with defining polynomial x^3 - x + 5 with a = 0.952080429567461? + 1.311248044077123?*I To: Complex Lazy Field Defn: a -> 0.952080429567461? + 1.311248044077123?*I Python This... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
variable name change map: From: Number Field in z with defining polynomial x^2 + 3 To: Number Field in a with defining polynomial x^2 + 3) sage: K. = QuadraticField(-3) sage: R. = K[] sage: D. = K.extension(y) sage: D_abs. = D.absolute_field() sage: D_abs.structure()0 -a Python subfield(alpha, name=None, names=None) [s... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
sage: CLF_from_K = K.coerce_embedding(); CLF_from_K Generic morphism: From: Cyclotomic Field of order 5 and degree 4 To: Complex Lazy Field Defn: z -> 0.309016994374948? + 0.951056516295154?*I sage: CLF_from_L = L.coerce_embedding(); CLF_from_L Generic morphism: From: Number Field in z0 with defining polynomial x^2 - 5... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
RIF(0.51, 0.52)) sage: K. = NumberField(poly, embedding=emb) sage: sqrt2 = -a^3 + 3*a sage: sqrt3 = -a^2 + 2 sage: assert sqrt2 ** 2 == 2 and sqrt3 ** 2 == 3 sage: L, elts, phi = K.subfield_from_elements([sqrt2, 1 - sqrt2/3]) sage: L Number Field in a0 with defining polynomial x^2 - 2 with a0 = 1.414213562373095? sage:... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
a number field P β a prime ideal of self others β either 'positive' (default), in which case the element will have nonnegative valuation at all other primes of self, or 'negative', in which case the element will have nonpositive valuation at all other primes of self Note When π is principal (e.g., always when self has... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
the βpositiveβ case. Use pari:idealappr with exponent of β 1 and invert the result in the βnegativeβ case. unit_group(proof=None) [source] Return the unit group (including torsion) of this number field. ALGORITHM: Uses PARIβs pari:bnfinit command. INPUT: proof β boolean (default: True); flag passed to PARI Note The gro... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4, -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1, a^16 - 2*a^15 - 2*a^13 - a^12 - a^11 - 2*a^10 + a^9 - 2*a^8 + 2*a^7 - 3*a^6 - 3*a^4 - 2*a^3 - a^2 - 4*a + 2, -a^15 - a^14... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
a^13 + a^12 + a^10 - a^7 - a^6 - a^2 - 1, 2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 - 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4, -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1, a^16 - 2*a^15 - 2*a^13 - a^12 - a^11 - 2*a^10 + a^9 - 2*a^8 +... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
extends uniquely: Sage sage: CyclotomicField(5).valuation(ZZ.valuation(5)) # needs sage.rings.padics 5-adic valuation Python When the extension is not unique, this does not work: Sage sage: K.valuation(ZZ.valuation(5)) # needs sage.rings.padics Traceback (most recent call last): ... ValueError: The valuation Gauss valu... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
See also Order.valuation(), pAdicGeneric.valuation() zeta(n=2, all=False) [source] Return one, or a list of all, primitive π -th root of unity in this field. INPUT: n β positive integer all β boolean; if False (default), return a primitive π -th root of unity in this field, or raise a ValueError exception if there ar... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
expensive, but we do use it if it is already known. EXAMPLES: Sage sage: x = polygen(QQ, 'x') sage: F. = NumberField(x^22 + 3) sage: F.zeta_order() 6 sage: F. = NumberField(x^2 - 7) sage: F.zeta_order() 2 Python sage.rings.number_field.number_field.NumberField_generic_v1(poly, name, latex_name, canonical_embedding=None... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
number 1 : Sage sage: len([d for d in range(2,200) ....: if not is_square(d) and QuadraticField(d,'a').class_number() == 1]) 121 Python discriminant(v=None) [source] Return the discriminant of the ring of integers of the number field, or if v is specified, the determinant of the trace pairing on the elements of the lis... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
= QuadraticField(-431) sage: K.class_number() 21 sage: K.hilbert_class_field_defining_polynomial(name='z') z^21 + 6*z^20 + 9*z^19 - 4*z^18 + 33*z^17 + 140*z^16 + 220*z^15 + 243*z^14 + 297*z^13 + 461*z^12 + 658*z^11 + 743*z^10 + 722*z^9 + 681*z^8 + 619*z^7 + 522*z^6 + 405*z^5 + 261*z^4 + 119*z^3 + 35*z^2 + 7*z + 1 Pytho... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
True) latex_name β latex variable name (default: π· ) OUTPUT: a number field defined by a quadratic polynomial. Unless otherwise specified, it has an embedding into π
or πΆ by sending the generator to the positive or upper-half-plane root. EXAMPLES: Sage sage: QuadraticField(3, 'a') Number Field in a with defining pol... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
in y with defining polynomial x^2 - 5 with y = 2.236067977499790? sage: K1 == K4 False Python sage.rings.number_field.number_field.is_AbsoluteNumberField(x) [source] Return True if x is an absolute number field. EXAMPLES: Sage sage: from sage.rings.number_field.number_field import is_AbsoluteNumberField sage: x = polyg... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
8, 12, 13] sage: [D for D in range(-15,15) ....: if not is_square(D) and QuadraticField(D,'a').disc() == D] [-15, -11, -8, -7, -4, -3, 5, 8, 12, 13] Python sage.rings.number_field.number_field.is_real_place(v) [source] Return True if π£ is real, False if π£ is complex. INPUT: v β an infinite place of self OUTPUT: A boo... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
list(embs); permuted_embs.remove(id); permuted_embs.append(id) sage: [e(a) for e in permuted_embs] # random - but natural map is not first [a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a] sage: permuted_embs != a True sage: from sage.rings.number_field.number_field import put_natural_embedding_first sage: put_na... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
Field in a with defining polynomial x^3 - 2 To: Complex Field with 53 bits of precision Defn: a |--> -0.62996052494743... - 1.09112363597172*I sage: e = refine_embedding(e,Infinity); e Ring morphism: From: Number Field in a with defining polynomial x^3 - 2 To: Algebraic Field Defn: a |--> -0.6299605249474365? - 1.09112... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
hilbert_symbol() hilbert_symbol_negative_at_S() is_absolute() logarithmic_embedding() minkowski_embedding() optimized_representation() optimized_subfields() order() places() real_places() relative_degree() relative_different() relative_discriminant() relative_polynomial() relative_vector_space() relativize() subfields(... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
Title: Hugging Face - The AI community building the future. URL Source: Warning: Target URL returned error 429: Too Many Requests Markdown Content: 429 --- We had to rate limit you. If you think it's an error, upgrade to a paid Enterprise Hub | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
Title: Solving Recurrence Relations URL Source: Markdown Content: Section 2.4 Solving Recurrence Relations ---------------------------------------- ΒΆ ###### Example 2.4.1 Find a recurrence relation and initial conditions for 1,5,17,53,161,485β¦. Solution Finding the recurrence relation would be easier if we had some co... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
\(a_0 = 1\text{.}\) We are going to try to _solve_ these recurrence relations. By this we mean something very similar to solving differential equations: we want to find a function of n (a closed formula) which satisfies the recurrence relation, as well as the initial condition.2 Recurrence relations are sometimes calle... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
Solve the recurrence relation a n=a nβ1+n with initial term a 0=4. Solution To get a feel for the recurrence relation, write out the first few terms of the sequence: \(4, 5, 7, 10, 14, 19, \ldots\text{.}\) Look at the difference between terms. \(a_1 - a_0 = 1\) and \(a_2 - a_1 = 2\) and so on. The key thing here is tha... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
of the form a n=a nβ1+f(n) where βk=1 n f(k) has a known closed formula. If you rewrite the recurrence relation as a nβa nβ1=f(n), and then add up all the different equations with n ranging between 1 and n, the left-hand side will always give you a nβa 0. The right-hand side will be βk=1 n f(k), which is why we need to... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
\(a_2\text{:}\) \begin{equation*} a_3 = ((a_0 + 1) + 2) + 3. \end{equation*} We notice a pattern. Each time, we take the previous term and add the current index. So \begin{equation*} a_n = ((((a_0 + 1) +2)+3)+\cdots + n-1) + n. \end{equation*} Regrouping terms, we notice that \(a_n\) is just \(a_0\) plus the sum of the... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
\cdot 3^{n-2} + \cdots + 2\cdot 3 + 2. \end{align*} Now we simplify. \(a_0 = 1\text{,}\) so we have \(3^n + \langle\text{stuff}\rangle\text{.}\) Note that all the other terms have a 2 in them. In fact, we have a geometric sum with first term \(2\) and common ratio \(3\text{.}\) We have seen how to simplify \(2 + 2\cdot... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
contain parts that look geometric. Perhaps the solution will take the form r n for some constant r. The nice thing is, we know how to check whether a formula is actually a solution to a recurrence relation: plug it in. What happens if we plug in r n into the recursion above? We get r nβr nβ1β6 r nβ2=0. Now solve for r:... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
Example 2.4.6 Solve the recurrence relation a n=7 a nβ1β10 a nβ2 with a 0=2 and a 1=3. Solution Rewrite the recurrence relation \(a_n - 7a_{n-1} + 10a_{n-2} = 0\text{.}\) Now form the characteristic equation: \begin{equation*} x^2 - 7x + 10 = 0 \end{equation*} and solve for \(x\text{:}\) \begin{equation*} (x - 2) (x - ... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
numbers (requiring the quadratic formula to find them). In these cases, we know what the solution to the recurrence relation looks like. However, it is possible for the characteristic polynomial to only have one root. This can happen if the characteristic polynomial factors as (xβr)2. It is still the case that r n woul... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
+ 3b. \end{align*} Since \(a = 1\text{,}\) we find that \(b = \frac{1}{3}\text{.}\) Therefore the solution to the recurrence relation is \begin{equation*} a_n = 3^n + \frac{1}{3}n3^n. \end{equation*} Although we will not consider examples more complicated than these, this characteristic root technique can be applied to... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
2^3\\ \vdots\amp \vdots \\ a_n - a_{n-1} \amp = 2^n \end{align*} which sums to \(a_n - a_0 = 2^{n+1} - 2\) (using the multiply-shift-subtract technique from Section 2.2 for the right-hand side). Substituting \(a_0 = 5\) and solving for \(a_n\) completes the solution. ###### 3 Show that 4 n is a solution to the recurren... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
16 Skittles, the fourth time 64 Skittles, etc. 1. Find both a recursive and closed formula for how many Skittles the _n_ th customer gets. 2. Check your solution for the closed formula by solving the recurrence relation using the Characteristic Root technique. ###### 10 You have access to 1Γ1 tiles which come in 2 diff... | {
"page_id": null,
"source": 6843,
"title": "from dpo"
} |
Title: 7.11 Expected Value - Contemporary Mathematics | OpenStax URL Source: Markdown Content: 7.11 Expected Value - Contemporary Mathematics | OpenStax =============== ![Image 2: Revisit consent button]( ![Image 3: Close]( ![Image 4: Brand logo]( This website utilizes technologies such as cookies to enable essential ... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
Duration 1 year * Description Customer support * Cookie _accounts_session_production * Duration 1 year 1 month 4 days * Description Cookies that are required for authentication and necessary OpenStax functions. * Cookie nudge_study_guides_page_counter * Duration 1 year 1 month 4 days * Description Product analytics * C... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
Duration session * Description This cookie is used by Elastic Load Balancing from Amazon Web Services to effectively balance load on the servers. Functional - [x] Functional cookies help perform certain functionalities like sharing the content of the website on social media platforms, collecting feedback, and other thi... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
embedded YouTube video. * Cookie yt-remote-fast-check-period * Duration session * Description The yt-remote-fast-check-period cookie is used by YouTube to store the user's video player preferences for embedded YouTube videos. Analytics - [x] Analytical cookies are used to understand how visitors interact with the websi... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
data includes the number of visitors, their source, and the pages they visit anonymously. * Cookie _fbp * Duration 3 months * Description Facebook sets this cookie to display advertisements when either on Facebook or on a digital platform powered by Facebook advertising after visiting the website. * Cookie _ga_* * Dura... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
on high-traffic sites. * Cookie pi_pageview_count * Duration Never Expires * Description Marketing automation tracking cookie * Cookie pulse_insights_udid * Duration Never Expires * Description User surveys * Cookie pi_visit_track * Duration Never Expires * Description Marketing cookie * Cookie pi_visit_count * Duratio... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
cookie to track and improve the performance of advertising campaigns, as well as to personalise the user experience. * Cookie _tt_enable_cookie * Duration 3 months * Description Tiktok set this cookie to collect data about behaviour and activities on the website and to measure the effectiveness of the advertising. * Co... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
the new or old player interface. * Cookie VISITOR_PRIVACY_METADATA * Duration 6 months * Description YouTube sets this cookie to store the user's cookie consent state for the current domain. * Cookie yt.innertube::requests * Duration Never Expires * Description YouTube sets this cookie to register a unique ID to store ... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
minute * Description No description available. * Cookie _ce.clock_data * Duration 1 day 1 minute * Description Description is currently not available. * Cookie cebsp_ * Duration session * Description Description is currently not available. * Cookie _swb_consent_ * Duration 7 days 1 minute * Description Description is c... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
Positional System]( 3. [4.2 Early Numeration Systems]( 4. [4.3 Converting with Base Systems]( 5. [4.4 Addition and Subtraction in Base Systems]( 6. [4.5 Multiplication and Division in Base Systems]( 7. Chapter Summary 1. [Key Terms]( 2. [Key Concepts]( 3. [Videos]( 4. [Projects]( 5. [Chapter Review]( 6. [Chapter Test](... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
The Binomial Distribution]( 12. [7.11 Expected Value]( 13. Chapter Summary 1. [Key Terms]( 2. [Key Concepts]( 3. [Formula Review]( 4. [Projects]( 5. [Chapter Review]( 6. [Chapter Test]( 9. 8 Statistics 1. [Introduction]( 2. [8.1 Gathering and Organizing Data]( 3. [8.2 Visualizing Data]( 4. [8.3 Mean, Median and Mode]( ... | {
"page_id": null,
"source": 6845,
"title": "from dpo"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.