Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
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Proposition 9.27. The bracket polynomial \( \langle D\rangle \) is invariant under Reidemeister moves (II) and (III). | Example. For the bridge and loop we get from (2) and (3),\n\n\[ \langle {D}_{b}^{ + }\rangle = - {A}^{-3},\;\langle {D}_{b}^{ - }\rangle = - {A}^{3},\;\langle {D}_{\ell }^{ + }\rangle = - {A}^{3},\;\langle {D}_{\ell }^{ - }\rangle = - {A}^{-3}. \]\n\n(4)\n\nLet us study next moves of type (I). With the notation there w... | No |
Theorem 9.28. The Kauffman polynomial \( {f}_{D}\left( A\right) = {\left( -{A}^{3}\right) }^{-w\left( D\right) }\langle D\rangle \) is an invariant of oriented diagrams. We can therefore uniquely define \( {f}_{L}\left( A\right) \) for an oriented link, by setting \( {f}_{L}\left( A\right) = {f}_{D}\left( A\right) \) f... | Proof. It is readily checked that the writhe is unchanged under the moves (II) and (III), hence \( {f}_{D}\left( A\right) \) is invariant under (II) and (III) by Proposition 9.27. As to Reidemeister (I), the figure shows that whatever way the curve in \( {D}^{\prime } \) is oriented we get a negative weight, whereas fo... | Yes |
Proposition 9.30. Let \( D \) be a positive alternating diagram with Tait graph \( G = \left( {V, E, F}\right) \) . Then\n\n\[ \langle D\rangle = {A}^{2\left| V\right| - \left| E\right| - 2}T\left( {G; - {A}^{-4}, - {A}^{4}}\right) ,\] \n\nwhere \( T\left( G\right) \) is the Tutte polynomial. The mirror image is given ... | Note that because of \( 2\left| V\right| - \left| E\right| - 2 = \left| V\right| - \left| F\right| \) (by Euler’s formula), we may also write\n\n\[ \langle D\rangle = {A}^{\left| V\right| - \left| F\right| }T\left( {G; - {A}^{-4}, - {A}^{4}}\right) ,\langle \overline{D}\rangle = {A}^{\left| {V}^{ * }\right| - \left| {F... | Yes |
Corollary 9.31. Let \( L \) be an alternating link, \( D \) a positive alternating diagram, and \( G = \left( {V, E, F}\right) \) the Tait graph. Then\n\n\[ \n{f}_{L}\left( A\right) = {\left( -1\right) }^{\left| E\right| }{A}^{\left| V\right| - \left| F\right| - {3w}\left( D\right) }T\left( {G; - {A}^{-4}, - {A}^{4}}\r... | Take \( A = t = 1 \) . We know from Corollary 9.24 that \( T\left( {G; - 1, - 1}\right) = \) \( {\left( -1\right) }^{\left| E\right| }{\left( -2\right) }^{c\left( q\right) - 1} \), where \( c\left( q\right) \) is the number of Eulerian cycles in the all-crossing transition system. But this is just the number \( c\left(... | Yes |
Lemma 10.1. Let \( G = \left( {V, E, F}\right) \) be a simple connected plane graph. Then there exists an orientation such that for any face \( f \) (except possibly the outer face) the number of clockwise oriented edges is odd. | Proof. We use induction on the number of faces. If \( \left| F\right| = 1 \), then all edges are bridges, and the assertion is vacuously satisfied. Suppose \( \left| F\right| > 1 \) . Then there exists a non-bridge \( e \) on the boundary of the outer face. Let \( f \) be the other face incident with \( e \) . In the g... | Yes |
Lemma 10.2. Let \( G = \left( {V, E, F}\right) \) be a simple connected plane graph without bridges, oriented according to the previous lemma, and let \( C \) be a circuit. Then the number of edges oriented clockwise in \( C \) has opposite parity to the number of vertices in the interior of \( C \) . | Proof. Let \( H = \left( {{V}^{\prime },{E}^{\prime },{F}^{\prime }}\right) \) be the subgraph of \( G \) consisting of the interior of \( C \), together with \( C \) . By Euler’s formula (disregarding the outer face), \[ \left| {V}^{\prime }\right| - \left| {E}^{\prime }\right| + \left| {F}^{\prime }\right| = 1 \] (3)... | Yes |
Theorem 10.3. Let \( G = \left( {V, E, F}\right) \) be a simple connected plane graph without bridges. Then the orientation given in Lemma 10.1 is Pfaffian. | Proof. Let \( A = \left( {a}_{ij}\right) \) be the oriented adjacency matrix of \( G \), and \( \sigma = {\sigma }_{1}\cdots {\sigma }_{t} \in {S}_{e} \) with \( {a}_{\sigma } \neq 0 \) . The cycles \( {\sigma }_{i} \) induce a partition of \( V \) into circuits \( {C}_{i} \) of even length and edges (if \( {\sigma }_{... | Yes |
Corollary 10.4. Let \( G \) be a simple connected plane graph without bridges and \( A \) its oriented adjacency matrix according to a Pfaffian orientation. Then\n\n\[ M\left( G\right) = \sqrt{\det A}. \] | Example. Consider the prism \( P \) embedded in the plane with the Pfaffian orientation given in the figure.\n\n\n\n\[ A = \left( \begin{array}{rrrrrr} 0 & 1 & 1 & 0 & - 1 & 0 \\ - 1 & 0 & 0 & 1 & 0 & 1 \\ - 1 & 0 & ... | No |
Theorem 10.6. For \( n \) even,\n\n\[ \n\left. {M\left( {m, n}\right) = \det \left( \begin{matrix} \left( \begin{aligned} m \\ 0 \end{aligned}\right) & - \left( \begin{aligned} {m - 1} \\ 1 \end{aligned}\right) & \left( \begin{aligned} {m - 2} \\ 2 \end{aligned}\right) & \ldots \\ 0 & \left( \begin{aligned} m \\ 0 \end... | Let us finally look at the case \( m = n \) . The resultant matrix in (14) is an \( n \times n \) -matrix. We perform three operations: First add row \( \frac{n}{2} + i \) to row \( i\left( {i = 1,\ldots ,\frac{n}{2}}\right) \), secondly factor out the 2’s, and thirdly subtract row \( i \) from row \( \frac{{n}^{ - }}{... | Yes |
Proposition 10.8. For \( G = \left( {V, E}\right) \) we have\n\na. \( \mathcal{E}\left( {G;z}\right) = {\left( 1 - z\right) }^{\left| E\right| - \left| V\right| + k\left( G\right) }{z}^{\left| V\right| - k\left( G\right) }T\left( {G;\frac{1}{z},\frac{1 + z}{1 - z}}\right) \) | Proof. We show that \( \mathcal{E}\left( z\right) \) is a chromatic invariant and use the Recipe Theorem 9.5. For a bridge or loop we have\n\n\[ \mathcal{E}\left( {\text{ bridge };z}\right) = 1,\;\mathcal{E}\left( {\text{ loop };z}\right) = 1 + z. \]\n\nIf \( e \) is a bridge of \( G \), then no Eulerian subgraph conta... | Yes |
Theorem 10.10. Let \( G = {L}_{m, n} \) and \( {G}^{t} \) the terminal graph with adjacency matrix \( A \) according to the orientation above. Then\n\n\[ \mathcal{E}\left( {G;z}\right) = \left| {\operatorname{Pf}\left( A\right) }\right| . \] | Proof. We number any \( {K}_{4} \) by\n\n\n\nand obtain for the Pfaffian\n\n\[ {a}_{12}{a}_{34} - {a}_{13}{a}_{24} + {a}_{14}{a}_{23} = 1 - 1 + 1 = 1. \]\n\nFor an Eulerian subgraph \( U \) of \( G \), denote by \( \... | Yes |
Lemma 10.11. Let \( M \) be a cyclic \( n \times n \) matrix, with \( a\left( k\right) \) and \( P \) as defined above. Then \( \widetilde{M} = {P}^{-1}{MP} \) is a diagonal block matrix, with\n\n\[ \n{\widetilde{m}}_{kk} = \lambda \left( \frac{2\pi k}{n}\right) ,\;k = 1,\ldots, n, \]\n\nwhere \( \lambda \) is the \( t... | Proof. We compute\n\n\[ \n{\widetilde{m}}_{k\ell } = \mathop{\sum }\limits_{{r, s = 1}}^{n}\overline{{p}_{kr}}{m}_{rs}{p}_{s\ell } \]\n\n\[ \n= \frac{1}{n}{I}_{t}\mathop{\sum }\limits_{{r = 1}}^{n}\exp \left( {-{kr}\frac{\pi i}{n}}\right) \mathop{\sum }\limits_{{s = 1}}^{n}a\left( {s - r}\right) \exp \left( {s\ell \fra... | Yes |
Lemma 10.15. Suppose the function \( g : \mathbb{N} \rightarrow \mathbb{R} \) satisfies \( g\left( {r + s}\right) \leq \) \( g\left( r\right) g\left( s\right) \) for all \( r \) and \( s \) . Then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}g{\left( n\right) }^{\frac{1}{n}} \) exists, and it is, in fact, equal to... | Proof. Fix \( r \) and \( \ell \) with \( \ell \leq r \) . The inequality implies by induction \( g\left( {\ell + {kr}}\right) \leq g\left( \ell \right) g{\left( r\right) }^{k} \), and thus\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\sup g{\left( \ell + kr\right) }^{\frac{1}{\ell + {kr}}} \leq g{\left( r\righ... | Yes |
Proposition 10.17. For all \( m \) , \[ {\Theta }_{m} = \mathop{\lim }\limits_{{n \rightarrow \infty }}f{\left( m, n\right) }^{\frac{1}{n}} = {\Lambda }_{m}. \] | Proof. Since \( \left| \lambda \right| \leq {\Lambda }_{m} \) for any eigenvalue \( \lambda \) of \( {T}_{m},{\Lambda }_{m}^{n} > 0 \) is the largest eigenvalue of \( {T}_{m}^{n} \) . Now by (8), \[ f\left( {m, n}\right) = {\mathbf{1}}^{T}{T}_{m}^{n}\mathbf{1} \leq {\Lambda }_{m}^{n}\left( {{\mathbf{1}}^{T}\mathbf{1}}\... | Yes |
Theorem 10.18. The limit \( \mathop{\lim }\limits_{{m, n \rightarrow \infty }}f{\left( m, n\right) }^{\frac{1}{mn}} \) exists, and we have\n\n\[ \xi = \mathop{\lim }\limits_{{m, n \rightarrow \infty }}f{\left( m, n\right) }^{\frac{1}{mn}} = \mathop{\lim }\limits_{{m \rightarrow \infty }}{\Lambda }_{m}^{\frac{1}{m}}. \]... | Proof. According to (11) it remains to show that \( \mathop{\lim }\limits_{{m \rightarrow \infty }}{\Lambda }_{m}^{\frac{1}{m}} \) exists, but this follows immediately from the lemma of Fekete. Indeed, \( f\left( {r + s, n}\right) \leq f\left( {r, n}\right) f\left( {s, n}\right) \) implies\n\n\[ f{\left( r + s, n\right... | Yes |
Theorem 10.20 (Baxter). The partition function \( {Z}_{n}\left( {{x}_{1},\ldots ,{x}_{n}}\right. \) , \( \left. {{y}_{1},\ldots ,{y}_{n};a}\right) \) is symmetric in the variables \( {x}_{i} \) and the \( {y}_{j} \) . | Proof. Consider rows \( i \) and \( i + 1 \) of the \( n \times n \) -grid of O-atoms. We insert an additional triangle at the left boundary as in the figure and color the cells appropriately. By the boundary condition the only possible color for the new cell is \( i \) .\n\n![8ae744ce-978d-4445-a0e7-6f200f1a4c8e_501_0... | Yes |
Lemma 2.1.1. Let \( G \) be a finitely generated torsion-free abelian group generated by \( {x}_{1},\ldots ,{x}_{n} \), and assume that \( G \) cannot be generated by fewer than \( n \) elements. Then there is no nontrivial relation \( \mathop{\sum }\limits_{{1 \leq i \leq n}}{a}_{i}{x}_{i} = 0 \) with \( {a}_{i} \in \... | Proof. Assume the contrary, and among all sets of \( n \) generators and all such relations on them, choose one for which \( \mathop{\sum }\limits_{{1 \leq i \leq n}}\left| {a}_{i}\right| \) is the smallest. We distinguish two cases:\n\n- If at least two of the \( {a}_{i} \) are nonzero, then permuting subscripts and c... | Yes |
Corollary 2.1.4. With the notation of the theorem, if we denote by \( \bar{x} \) the class of an element of \( G \) in \( G/H \), we have\n\n\[ G/H = {\bigoplus }_{1 \leq i \leq r}\left( {\mathbb{Z}/{m}_{i}\mathbb{Z}}\right) \overline{{x}_{i}} \oplus {\bigoplus }_{r < i \leq n}\mathbb{Z}\overline{{x}_{i}}. \] | Proof. Clear. Note that this is an equality, not only an isomorphism. | No |
Corollary 2.1.5. Any subgroup of a finitely generated free abelian group is a finitely generated free abelian group of lower dimension. | Proof. Also clear, \( H \) being free on the \( {m}_{i}{x}_{i} \) for \( 1 \leq i \leq r \) and \( r \leq n \) . | No |
Corollary 2.1.6. Let \( V \in {\mathbb{Z}}^{n} \) be a column vector of \( n \) globally coprime integers. There exists an integral matrix \( A \in {\mathrm{{GL}}}_{n}\left( \mathbb{Z}\right) \) (in other words with determinant \( \pm 1 \) ) having \( V \) as first column. | Proof. In the proposition, we let \( G = {\mathbb{Z}}^{n} \) and \( H = \mathbb{Z}V \) . There exists a basis \( {A}_{1},\ldots ,{A}_{n} \) of \( G \) and \( d \in {\mathbb{Z}}_{ \geq 1} \) such that \( d{A}_{1} \) is a basis of \( H \) . In particular, \( V \in {dk}{A}_{1} \) for some \( k \in \mathbb{Z} \), and since... | Yes |
Theorem 2.1.7 (Elementary divisor theorem II). Let \( G \) be a finitely generated abelian group. There exist elements \( {x}_{1},\ldots ,{x}_{n} \) of \( G \) and positive integers \( {m}_{1},\ldots ,{m}_{r} \) for some \( r \leq n \) such that \( {m}_{i} > 1 \) for \( 1 \leq i \leq r,{m}_{i} \mid {m}_{i + 1} \) for \... | Proof. To prove existence, let \( {y}_{1},\ldots ,{y}_{N} \) be any generators of \( G \), and let \( {G}^{ * } = {\bigoplus }_{1 \leq i \leq N}\mathbb{Z}{Y}_{i} \simeq {\mathbb{Z}}^{N} \) be the free abelian group on \( N \) generators \( {Y}_{i} \) . There is a natural surjection from \( {G}^{ * } \) to \( G \) sendi... | Yes |
Corollary 2.1.8. Any subgroup of a finitely generated abelian group is finitely generated. | Proof. Once again, we use a finitely generated free abelian group \( {G}^{ * } \) and a surjective map from \( {G}^{ * } \) to \( G \) . If \( H \) is a subgroup of \( G \), denote by \( {H}^{ * } \) the inverse image of \( H \) by this map. By Corollary 2.1.5, \( {H}^{ * } \) is finitely generated, and the images of a... | Yes |
Theorem 2.1.9. Let \( G \) be a finite abelian group. There exist unique integers \( {m}_{i} > 1 \) for \( 1 \leq i \leq k \) such that \( {m}_{i} \mid {m}_{i + 1} \) for \( 1 \leq i < k \), and nonunique elements \( {g}_{i} \in G \) such that\n\n\[ G = {\bigoplus }_{1 \leq i \leq k}\left( {\mathbb{Z}/{m}_{i}\mathbb{Z}... | Proof. Indeed, if \( G \) is finite it is finitely generated. We have seen above as a consequence of Theorem 2.1.7 that any such group can be written\n\n\[ G = {\bigoplus }_{1 \leq i \leq r}\left( {\mathbb{Z}/{m}_{i}\mathbb{Z}}\right) {g}_{i} \oplus {\bigoplus }_{r < i \leq k}\mathbb{Z}{g}_{i}, \]\n\nfor some \( {g}_{i... | Yes |
Theorem 2.1.10 (Smith normal form). Let \( A \) be a square integral matrix with nonzero determinant. There exist two unimodular matrices \( U \) and \( V \) and a diagonal integral matrix \( D \) with strictly positive diagonal entries such that \( D = {UAV} \), and if \( D = \left( {d}_{i, j}\right) \) then \( {d}_{i... | Proof. We apply Theorem 2.1.3 to \( G = {\mathbb{Z}}^{k} \) and \( H \) the group of \( \mathbb{Z} \) -linear combinations of columns of \( A \) considered as elements of \( {\mathbb{Z}}^{k} \) . We leave to the reader to check that we thus obtain the present theorem (Exercise 2). | No |
Proposition 2.1.12. Let \( G \) be an abelian group and let \( g \in G \) be an element of finite order \( k = {k}_{1}{k}_{2} \) with \( {k}_{1} \) and \( {k}_{2} \) coprime. There exist \( {g}_{1} \) and \( {g}_{2} \) in \( G \) of respective orders \( {k}_{1} \) and \( {k}_{2} \) such that \( g = {g}_{1}{g}_{2} \) . | Proof. Since \( {k}_{1} \) and \( {k}_{2} \) are coprime there exist integers \( {u}_{1} \) and \( {u}_{2} \) such that \( {u}_{1}{k}_{1} + {u}_{2}{k}_{2} = 1 \) . We set \( {g}_{1} = {g}^{{u}_{2}{k}_{2}} \) and \( {g}_{2} = {g}^{{u}_{1}{k}_{1}} \) . It is clear that \( {g}_{1}{g}_{2} = g \), and furthermore by definit... | Yes |
Theorem 2.1.14. Let \( G \) be a finite abelian group, and using the notation of Theorem 2.1.9, write\n\n\[ G = {\bigoplus }_{1 \leq i \leq k}\left( {\mathbb{Z}/{m}_{i}\mathbb{Z}}\right) {g}_{i} \]\n\nDenote by \( D \) be the \( k \times k \) diagonal matrix whose diagonal entries are the integers \( {m}_{i} \) and by ... | Proof. By definition, the following sequence is exact:\n\n\[ 1 \rightarrow {\bigoplus }_{i = 1}^{k}{m}_{i}\mathbb{Z} \rightarrow {\mathbb{Z}}^{k}\overset{\phi }{ \rightarrow }G \rightarrow 1 \]\n\nwhere\n\n\[ \phi \left( {{x}_{1},\ldots ,{x}_{k}}\right) = \mathop{\prod }\limits_{{1 \leq i \leq k}}{x}_{i}{g}_{i}. \]\n\n... | Yes |
Proposition 2.1.16. Let \( G \) be a finite abelian group. The dual group \( \widehat{G} \) is noncanonically isomorphic to \( G \) (hence has the same cardinality). | Proof. By the structure theorem for finite abelian groups (Theorem 2.1.9) we know that\n\n\[ G = {\bigoplus }_{1 \leq i \leq k}\left( {\mathbb{Z}/{m}_{i}\mathbb{Z}}\right) {g}_{i} \simeq {\bigoplus }_{1 \leq i \leq k}\left( {\mathbb{Z}/{m}_{i}\mathbb{Z}}\right) \]\n\nfor certain integers \( {m}_{i} \) and \( {g}_{i} \i... | Yes |
Corollary 2.1.17. Let \( G \) be a finite abelian group and \( H \) a subgroup of \( G \) . Any character of \( H \) can be extended to exactly \( \left\lbrack {G : H}\right\rbrack \) characters of \( G \) . In particular, the natural restriction map from \( \widehat{G} \) to \( \widehat{H} \) is surjective. | Proof. Let \( f \) be the above restriction map. The kernel of \( f \) is the group of characters \( \chi \) of \( G \) that are trivial on \( H \), in other words the characters of \( G/H \) . It follows by the proposition that the cardinality of the image of \( f \) is equal to\n\n\[ \frac{\left| \widehat{G}\right| }... | Yes |
Corollary 2.1.18. If \( g \) is not the unit element of \( G \) there exists \( \chi \in \widehat{G} \) such that \( \chi \left( g\right) \neq 1 \) . | Proof. If \( n > 1 \) is the order of \( g \) we set \( \chi \left( {g}^{k}\right) = {\zeta }_{n}^{k} \), which defines a character such that \( \chi \left( g\right) \neq 1 \) on the subgroup \( H \) of \( G \) generated by \( g \), and we extend \( \chi \) to \( G \) using the above corollary. | Yes |
Corollary 2.1.19. The natural map \( a \mapsto \left( {\chi \mapsto \chi \left( a\right) }\right) \) gives a canonical isomorphism from \( G \) to the dual of its dual. | Proof. By the preceding corollary this map is injective, and since both groups have the same cardinality it is an isomorphism. | No |
Proposition 2.1.20. Let \( G \) be a finite abelian group and let \( K \) be a commutative field.\n\n(1) If \( {\chi }_{1} \) and \( {\chi }_{2} \) are distinct group homomorphisms from \( G \) to \( {K}^{ * } \) then\n\n\[ \mathop{\sum }\limits_{{g \in G}}{\chi }_{1}\left( g\right) {\chi }_{2}^{-1}\left( g\right) = 0 ... | Proof. The two statements of (1) are clearly equivalent, as are those of (2). Set \( S = \mathop{\sum }\limits_{{g \in G}}\chi \left( g\right) \) . Let \( h \in G \) such that \( \chi \left( h\right) \neq 1 \) . Then\n\n\[ \chi \left( h\right) S = \mathop{\sum }\limits_{{g \in G}}\chi \left( h\right) \chi \left( g\righ... | Yes |
Lemma 2.1.22. Let \( p \) be a prime number, \( s \) an integer such that \( s \equiv 1 \) \( \left( {\;\operatorname{mod}\;p}\right) \), and let \( n \in {\mathbb{Z}}_{ > 0} \) . When \( p = 2 \), assume that either \( s \equiv 1\left( {\;\operatorname{mod}\;4}\right) \) or \( n \) is odd. Then\n\n\[ \n{v}_{p}\left( {... | Proof. Write \( n = {p}^{v}m \) with \( p \nmid m \) . We prove the lemma by induction on \( v \) . Assume first that \( v = 0 \), so that \( n = m \) . By the binomial theorem, we have\n\n\[ \n{s}^{m} - 1 = \mathop{\sum }\limits_{{1 \leq k \leq m}}\left( \begin{matrix} m \\ k \end{matrix}\right) {\left( s - 1\right) }... | Yes |
Corollary 2.1.23. If \( p = 2 \) and \( s \equiv 1\left( {\;\operatorname{mod}\;2}\right) \) then for all \( n \in {\mathbb{Z}}_{ > 0} \) we have\n\n\[ \n{v}_{p}\left( {{s}^{n} - 1}\right) = \left\{ \begin{array}{ll} {v}_{p}\left( {s - 1}\right) & \text{ if }n\text{ is odd,} \\ {v}_{p}\left( {{s}^{2} - 1}\right) + {v}_... | Proof. The case \( n \) odd is given by the lemma. When \( n \) is even, since \( {s}^{2} \equiv 1 \) \( \left( {\;\operatorname{mod}\;4}\right) \) the lemma gives \( {v}_{p}\left( {{s}^{n} - 1}\right) = {v}_{p}\left( {{\left( {s}^{2}\right) }^{n/2} - 1}\right) = {v}_{p}\left( {{s}^{2} - 1}\right) + {v}_{p}\left( {n/2}... | Yes |
Proposition 2.1.24. Let \( m \geq 2 \) be an integer, and let \( m = \mathop{\prod }\limits_{{1 \leq i \leq g}}{p}_{i}^{{v}_{i}} \) be its decomposition into a product of powers of distinct primes. The abelian group structure of \( {\left( \mathbb{Z}/m\mathbb{Z}\right) }^{ * } \) is given as follows:\n\n(1) We have\n\n... | Proof. (1). I first claim that if \( m = {m}_{1}{m}_{2} \) with \( \gcd \left( {{m}_{1},{m}_{2}}\right) = 1 \), then \( {\left( \mathbb{Z}/m\mathbb{Z}\right) }^{ * } \simeq {\left( \mathbb{Z}/{m}_{1}\mathbb{Z}\right) }^{ * } \times {\left( \mathbb{Z}/{m}_{2}\mathbb{Z}\right) }^{ * } \) . Indeed, there exist integers \(... | Yes |
Corollary 2.1.25. For \( m \geq 2 \), the group \( {\left( \mathbb{Z}/m\mathbb{Z}\right) }^{ * } \) is cyclic if and only if \( m = 2,4,{p}^{k} \), or \( 2{p}^{k} \) for \( p \) an odd prime and \( k \geq 1 \) . | Proof. Note that in a cyclic group the number of elements of order dividing 2 is less than or equal to 2 . From the proposition it follows that the number of such elements is exactly equal to \( {2}^{{\omega }_{o}\left( m\right) + {\omega }_{2}\left( m\right) } \), where \( {\omega }_{o}\left( m\right) \) is the number... | Yes |
Lemma 2.1.26. Let \( p \) be an odd prime, let \( v \geq 2 \), and let \( g \) be a primitive root modulo \( {p}^{v} \) . For any a coprime to \( p \) there exist \( x \) and \( y \) such that\n\n\[ a \equiv {g}^{{p}^{v - 1}x}{\left( 1 + p\right) }^{y}\left( {\;\operatorname{mod}\;{p}^{v}}\right) ,\]\n\nand \( x \) is ... | Proof. Since \( g \) is a primitive root, we can write \( a \equiv {g}^{x}\left( {\;\operatorname{mod}\;{p}^{v}}\right) \), so that \( {a}^{{p}^{v - 1}} \equiv {g}^{{p}^{v - 1}x}\left( {\;\operatorname{mod}\;{p}^{v}}\right) \), and since \( g \) has order \( {p}^{v - 1}\left( {p - 1}\right) \), it is clear that \( x \)... | Yes |
Proposition 2.1.29. The number of primitive characters modulo \( m \) is equal to \( q\left( m\right) \), where\n\n\[ q\left( m\right) = m\mathop{\prod }\limits_{{p\parallel m}}\left( {1 - \frac{2}{p}}\right) \mathop{\prod }\limits_{{{p}^{2} \mid m}}{\left( 1 - \frac{1}{p}\right) }^{2}. \] | Proof. I refer the reader to Section 10.1 for the elementary techniques used here. For any integer \( f \) denote by \( q\left( f\right) \) the number of primitive characters modulo \( f \) . By definition we have\n\n\[ \phi \left( m\right) = \left| {\left( \mathbb{Z}/m\mathbb{Z}\right) }^{ * }\right| = \left| {\left( ... | No |
Corollary 2.1.30. Let \( \chi \) be a primitive character modulo \( m \) with \( m \) even. Then for all \( n \) we have \( \chi \left( {n + m/2}\right) = - \chi \left( n\right) \) . | Proof. By the proposition we know that \( 4 \mid m \) . The result is thus trivial if \( n \) is even since both sides vanish; otherwise, denoting by \( {n}^{-1} \) an inverse of \( n \) modulo \( m \) we have since \( n \) is odd\n\n\[ \chi \left( {n + m/2}\right) = \chi \left( n\right) \chi \left( {1 + \left( {m/2}\r... | Yes |
Lemma 2.1.31. If \( \gcd \left( {a, b, c}\right) = 1 \) there exists an integer \( k \) such that\n\n\[ \gcd \left( {a + {kb}, c}\right) = 1. \] | Proof. Note that this lemma would immediately follow from Dirichlet's theorem on primes in arithmetic progression (see Theorem 10.5.30), but it is not necessary to use such a powerful tool. In fact, we can give \( k \) explicitly: I claim that\n\n\[ k = \mathop{\prod }\limits_{\substack{{p \mid c} \\ {p \nmid \left( {a... | Yes |
Lemma 2.1.32. Let \( \chi \) be a character modulo \( m \) and let \( d \mid m \) . Then \( \chi \) can be defined modulo \( d \) if and only if for all \( a \) such that \( a \equiv 1\left( {\;\operatorname{mod}\;d}\right) \) and \( \gcd \left( {a, m}\right) = 1 \) we have \( \chi \left( a\right) = 1 \) . | Proof. The condition is clearly necessary: if \( \chi \left( a\right) = {\chi }_{d}\left( a\right) \) for all \( a \) such that \( \gcd \left( {a, m}\right) = 1 \), then if in addition \( a \equiv 1\left( {\;\operatorname{mod}\;d}\right) \) we have \( \chi \left( a\right) = 1 \) . Conversely, assume the condition satis... | Yes |
Corollary 2.1.33. Let \( \chi \) be a character modulo \( m \), let \( d \mid m \) with \( d < m \) , and assume that \( \chi \) cannot be defined modulo \( d \) .\n\n(1) For all \( r \) we have\n\n\[ \mathop{\sum }\limits_{\substack{{a{\;\operatorname{mod}\;m}} \\ {a \equiv r\left( {\;\operatorname{mod}\;d}\right) } }... | Proof. The proof of (1) is identical to that of Proposition 2.1.20: by the lemma, there exists \( b \equiv 1\left( {\;\operatorname{mod}\;d}\right) \) with \( \gcd \left( {b, m}\right) = 1 \) and such that \( \chi \left( b\right) \neq \) 1. The map \( a \mapsto {ab} \) is clearly a bijection from the set of integers mo... | Yes |
Proposition 2.1.34. Let \( m \in {\mathbb{Z}}_{ \geq 1} \), let \( {m}_{1} \) and \( {m}_{2} \) be two coprime positive integers such that \( m = {m}_{1}{m}_{2} \), and let \( \chi \) be a Dirichlet character modulo \( m \) . (1) There exist unique characters \( {\chi }_{i} \) modulo \( {m}_{i} \) such that \( \chi = {... | Proof. (1). Since the \( {m}_{i} \) are coprime, there exist integers \( u \) and \( v \) such that \( u{m}_{1} + v{m}_{2} = 1 \) . In view of the map \( {f}_{2} \) defined in the proof of Proposition 2.1.24 (1), it is natural to set \( {\chi }_{1}\left( x\right) = \chi \left( {x{m}_{2}v + {m}_{1}u}\right) \) and \( {\... | Yes |
Corollary 2.1.35. Let \( m = \mathop{\prod }\limits_{p}{p}^{{v}_{p}\left( m\right) } \) with \( {v}_{p}\left( m\right) \geq 1 \) be the decomposition into prime powers of \( m \in {\mathbb{Z}}_{ \geq 1} \) . The order of any primitive character modulo \( m \) is divisible by \( h\left( m\right) = \mathop{\prod }\limits... | Proof. Let \( \chi \) be a primitive character modulo \( m \) . By the above proposition applied inductively we can write \( \chi = \mathop{\prod }\limits_{p}{\chi }_{p} \), where \( {\chi }_{p} \) is a primitive character modulo \( {p}^{{v}_{p}\left( m\right) } \), and the order of \( \chi \) will be equal to the LCM ... | Yes |
Proposition 2.1.36. We have\n\n\[ \mathop{\sum }\limits_{{a{\;\operatorname{mod}\;m}}}\chi \left( a\right) = \left\{ \begin{array}{ll} \phi \left( m\right) & \text{ if }\chi \text{ is the trivial character modulo }m, \\ 0 & \text{ otherwise. } \end{array}\right. \] | Dually, if \( a \in \mathbb{Z} \) is such that \( \gcd \left( {a, m}\right) = 1 \) then\n\n\[ \mathop{\sum }\limits_{{\chi {\;\operatorname{mod}\;m}}}\chi \left( a\right) = \left\{ \begin{array}{ll} \phi \left( m\right) & \text{ if }a \equiv 1\left( {\;\operatorname{mod}\;m}\right) , \\ 0 & \text{ otherwise. } \end{arr... | No |
Corollary 2.1.37. Let \( \chi \) be a nontrivial character modulo \( m \), let \( I = \lbrack 1, m - 1]\), and let\n\n\[ S = \mathop{\sum }\limits_{{a \in I, a\text{ even }}}\chi \left( a\right) = - \mathop{\sum }\limits_{{a \in I, a\text{ odd }}}\chi \left( a\right) .\n\]\n\n(1) If either \( m \) is even or \( \chi \)... | Proof. (1). The fact that the two sums given in the corollary are opposite follows of course from Proposition 2.1.36, and if \( m \) is even then \( \chi \left( a\right) = 0 \) for\n\nall even \( a \), hence \( S = 0 \) trivially. On the other hand, if \( m \) is odd and \( \chi \) is even we have\n\n\[ S = \mathop{\su... | Yes |
Proposition 2.1.39. If \( \gcd \left( {a, m}\right) = 1 \) we have\n\n\[ \tau \left( {\chi, a}\right) = \overline{\chi \left( a\right) }\tau \left( \chi \right) . \] | Proof. Since \( \gcd \left( {a, m}\right) = 1 \), the map multiplication by \( a \) is a bijection of \( {\left( \mathbb{Z}/m\mathbb{Z}\right) }^{ * } \) to itself; hence setting \( y = {ax} \) we have\n\n\[ \tau \left( {\chi, a}\right) = \mathop{\sum }\limits_{{y{\;\operatorname{mod}\;m}}}\chi \left( {y{a}^{-1}}\right... | Yes |
Proposition 2.1.40. Let \( d = \gcd \left( {a, m}\right) \) and assume that \( \chi \) cannot be defined modulo \( m/d \) . Then \( \tau \left( {\chi, a}\right) = 0 \) . | Proof. Since \( \chi \) cannot be defined modulo \( m/d \), by Lemma 2.1.32 we can find \( b \) such that \( b \equiv 1\left( {{\;\operatorname{mod}\;m}/d}\right) ,\gcd \left( {b, m}\right) = 1 \), and \( \chi \left( b\right) \neq 1 \) . Thus\n\n\[ \chi \left( b\right) \tau \left( {\chi, a}\right) = \mathop{\sum }\limi... | Yes |
Corollary 2.1.41. If \( \chi \) is a nontrivial character modulo \( m \), then\n\n\[ \mathop{\sum }\limits_{{x{\;\operatorname{mod}\;m}}}\chi \left( x\right) = 0 \] | Proof. Apply the above proposition to \( a = 0 \) . | No |
Corollary 2.1.42. Assume that \( \chi \) is a primitive character. For all a (not necessarily prime to \( m \) ) we have\n\n\[ \tau \left( {\chi, a}\right) = \overline{\chi \left( a\right) }\tau \left( \chi \right) . \] | Proof. If \( \gcd \left( {a, m}\right) = 1 \) this is Proposition 2.1.39, and if \( d = \gcd \left( {a, m}\right) > 1 \) , then \( \chi \) cannot be defined modulo \( m/d \) so the result follows from Proposition 2.1.40. | Yes |
Corollary 2.1.43. Assume that \( \chi \) is a primitive character modulo \( m \) and let \( n = {km} \) be a multiple of \( m \) . Then\n\n\[ \mathop{\sum }\limits_{{x{\;\operatorname{mod}\;n}}}\chi \left( x\right) {\zeta }_{n}^{ax} = \left\{ \begin{array}{ll} 0 & \text{ if }k \nmid a \\ k\bar{\chi }\left( {a/k}\right)... | Proof. Immediate by writing \( x = {mq} + r \) with \( r{\;\operatorname{mod}\;m} \) and \( q{\;\operatorname{mod}\;k} \) and left to the reader. | No |
Proposition 2.1.45. If \( \chi \) is a primitive character modulo \( m \) then \( \left| {\tau \left( \chi \right) }\right| = \) \( {m}^{1/2} \) . | Proof. We have \( \overline{\tau \left( \chi \right) } = \mathop{\sum }\limits_{{a{\;\operatorname{mod}\;m}}}\overline{\chi \left( a\right) }{\zeta }_{m}^{-a} \) ; hence multiplying by \( \tau \left( \chi \right) \) and applying the above corollary we obtain\n\n\[{\left| \tau \left( \chi \right) \right| }^{2} = \mathop... | Yes |
Corollary 2.1.46. Let \( \chi \) be a not necessarily primitive character modulo \( m \), and let \( f \) be its conductor. Then \( \left| {\tau \left( \chi \right) }\right| = {f}^{1/2} \) if \( m/f \) is squarefree and coprime to \( f \) ; otherwise \( \tau \left( \chi \right) = 0 \) . | Proof. This follows from the above proposition and the formula \( \tau \left( \chi \right) = \) \( \mu \left( {m/f}\right) {\chi }_{f}\left( {m/f}\right) \tau \left( {\chi }_{f}\right) \) proved in Exercise 12. | No |
Corollary 2.1.47. If \( \chi \) is a primitive character we have\n\n\[ \tau \left( \chi \right) \tau \left( \bar{\chi }\right) = \chi \left( {-1}\right) m. \] | Proof. Indeed, by Proposition 2.1.39 we have\n\n\[ \overline{\tau \left( \chi \right) } = \mathop{\sum }\limits_{{x{\;\operatorname{mod}\;m}}}\bar{\chi }\left( x\right) {\zeta }_{m}^{-x} = \chi \left( {-1}\right) \tau \left( \bar{\chi }\right) ,\]\n\nso multiplying by \( \tau \left( \chi \right) \) and using Propositio... | Yes |
Proposition 2.2.1. (1) The symbol \( \\left( \\frac{a}{p}\\right) \) is a real primitive character modulo \( p \), and in particular \( \\left( \\frac{ab}{p}\\right) = \\left( \\frac{a}{p}\\right) \\left( \\frac{b}{p}\\right) \) . | Proof. By Corollary 2.4.3 below we know that \( \\left( \\mathbb{Z}/p\\mathbb{Z}\\right) ^{ * } \) is cyclic (of order \( p - 1) \) . Let \( g \\in \\mathbb{Z} \) be such that the class of \( g \) modulo \( p \) is a generator. Then if \( a \\in \\mathbb{Z} \) is coprime to \( p \), there exists an exponent \( k \) uni... | Yes |
Lemma 2.2.2. Let \( \chi \) be a real primitive character modulo \( m \), and let \( p \) be an odd prime. Then\n\n\[ \chi \left( p\right) = \left( \frac{\chi \left( {-1}\right) m}{p}\right) . \] | Proof. Let \( R = \mathbb{Z}\left\lbrack {\zeta }_{m}\right\rbrack \), which is a ring and a finitely generated free \( \mathbb{Z} \) - module since \( {\zeta }_{m} \) is an algebraic integer (in fact of degree \( \phi \left( m\right) \), but we do not need this). We do not need to know that in fact \( R \) is the ring... | Yes |
Corollary 2.2.3 (The basic quadratic reciprocity law). (1) If \( p \) and \( q \) are distinct odd primes, we have\n\n\[ \left( \frac{p}{q}\right) \left( \frac{q}{p}\right) = {\left( -1\right) }^{\left( {p - 1}\right) \left( {q - 1}\right) /4}. \]\n\n(2) If \( p \) is an odd prime, we have the two so-called complementa... | Proof. Set \( \chi \left( n\right) = \left( \frac{n}{q}\right) \), which is a real primitive character modulo \( q \) . We have \( \chi \left( {-1}\right) = {\left( -1\right) }^{\left( {q - 1}\right) /2} \), hence the above lemma gives\n\n\[ \left( \frac{p}{q}\right) = \left( \frac{{\left( -1\right) }^{\left( {q - 1}\r... | Yes |
Proposition 2.2.4. If \( m \) and \( n \) are coprime positive odd integers, the same quadratic reciprocity formula holds:\n\n\[ \left( \frac{m}{n}\right) \left( \frac{n}{m}\right) = {\left( -1\right) }^{\left( {n - 1}\right) \left( {m - 1}\right) /4}. \] | Proof. We note that if \( {n}_{1} \) and \( {n}_{2} \) are odd, then\n\n\[ {n}_{1}{n}_{2} - 1 = \left( {{n}_{1} - 1}\right) + \left( {{n}_{2} - 1}\right) + \left( {{n}_{1} - 1}\right) \left( {{n}_{2} - 1}\right) \equiv \left( {{n}_{1} - 1}\right) + \left( {{n}_{2} - 1}\right) \left( {\;\operatorname{mod}\;4}\right) .\n... | Yes |
Proposition 2.2.6. (1) For two nonzero integers \( m \) and \( n \) write \( m = \) \( {2}^{{v}_{2}\left( m\right) }{m}_{1} \) and \( n = {2}^{{v}_{2}\left( n\right) }{n}_{1} \) with \( {m}_{1} \) and \( {n}_{1} \) odd. Then\n\n\[ \left( \frac{n}{m}\right) = {\left( -1\right) }^{\left( {\left( {{m}_{1} - 1}\right) \lef... | Proof. We may assume that either \( m \) or \( n \) is odd; otherwise they are not coprime and the result is trivial. Since \( \left( \frac{2}{a}\right) = \left( \frac{a}{2}\right) \), it is clear that statement (1) follows from Proposition 2.2 .4 for \( m, n \) both positive. Using the definition of \( \left( \frac{m}... | Yes |
Lemma 2.2.7. If \( m \) is odd, for any \( k \) we have\n\n\[ \left( \frac{a + {km}}{m}\right) = {\left( -1\right) }^{\left( {\operatorname{sign}\left( m\right) - 1}\right) \left( {\operatorname{sign}\left( {a + {km}}\right) - \operatorname{sign}\left( a\right) }\right) /4}\left( \frac{a}{m}\right) . \] | Proof. When \( m > 0 \), we have periodicity because of the periodicity of the Legendre symbol. If \( m < 0 \), by definition of the Kronecker symbol we have\n\n\[ \left( \frac{a + {km}}{m}\right) = \operatorname{sign}\left( {a + {km}}\right) \left( \frac{a + {km}}{\left| m\right| }\right) \]\n\n\[ = \operatorname{sign... | Yes |
Lemma 2.2.8. If \( m \) is odd, then writing \( n = {2}^{{v}_{2}\left( n\right) }{n}_{1} \) and \( n + {km} = \) \( {2}^{{v}_{2}\left( {n + {km}}\right) }{\left( n + km\right) }_{1} \), we have\n\n\[ \left( \frac{m}{n + {km}}\right) = {\left( -1\right) }^{\left( {m - 1}\right) \left( {{\left( n + km\right) }_{1} - {n}_... | Proof. By Proposition 2.2.6 and the above lemma, we have\n\n\[ \left( \frac{m}{n + {km}}\right) = {\left( -1\right) }^{\left( {m - 1}\right) \left( {{\left( n + km\right) }_{1} - 1}\right) /4 + \left( {\operatorname{sign}\left( m\right) - 1}\right) \left( {\operatorname{sign}\left( {n + {km}}\right) - 1}\right) /4}\lef... | Yes |
Theorem 2.2.9. If \( m \equiv 0 \) or \( 1 \) modulo 4 is fixed, the Kronecker symbol \( \left( \frac{m}{n}\right) \) is periodic of period dividing \( \left| m\right| \) ; in other words for all \( k, n \) we have\n\n\[ \left( \frac{m}{n + {km}}\right) = \left( \frac{m}{n}\right) . \] | Proof. If \( m \equiv 1\left( {\;\operatorname{mod}\;4}\right) \) the result follows from the above lemma since in that case \( \left( {m - 1}\right) \left( {{\left( n + km\right) }_{1} - {n}_{1}}\right) \equiv 0\left( {\;\operatorname{mod}\;8}\right) \) . So assume \( m \equiv 0\left( {\;\operatorname{mod}\;4}\right) ... | Yes |
Proposition 2.2.12. The quantity \( {f}_{H}\left( {a, r}\right) \) does not depend on the half-system \( H \) . | Proof. Let \( {H}^{\prime } \) be another half-system. Then for any \( j \in H \) there exists \( \eta \left( j\right) = \pm 1 \) such that \( j = \eta \left( j\right) \pi \left( j\right) \), where \( \pi \) is a (necessarily bijective) map from \( H \) to \( {H}^{\prime } \) . For simplicity of notation, set \( {\sigm... | Yes |
Corollary 2.2.14. Let \( r \) be an odd positive integer and let a be any integer coprime to \( r \) . Then\n\n(1)\n\n\[{\left( -1\right) }^{S\left( {a, r}\right) } = \left\{ \begin{array}{ll} \left( \frac{a}{r}\right) & \text{ when }a\text{ is odd,} \\ \left( \frac{2a}{r}\right) & \text{ when }a\text{ is even. } \end{... | Proof. (1). Choose as half-system \( H \) modulo \( r \) the integers from 1 to \( \left( {r - 1}\right) /2 \) and keep the above notation. In particular, multiplication by \( a \) defines a function \( {\varepsilon }_{H}\left( j\right) \) with values \( \pm 1 \) and a permutation \( {\sigma }_{H} \) of \( H \) . For \... | No |
If \( D \) is a fundamental discriminant, the Kronecker symbol \( \left( \frac{D}{n}\right) \) defines a real primitive character modulo \( m = \left| D\right| \) . Conversely, if \( \chi \) is a real primitive character modulo \( m \) then \( D = \chi \left( {-1}\right) m \) is a fundamental discriminant \( D \) and \... | Proof. The definition of the Kronecker symbol and Theorem 2.2.9 show that \( \left( \frac{D}{n}\right) \) is a character modulo \( \left| D\right| \) . To show that it is primitive, it is sufficient to show that for any prime \( p \mid D \) it cannot be defined modulo \( D/p \) . Assume first that \( p \neq 2 \), and l... | Yes |
Proposition 2.2.16 (Poisson summation formula). Let \( f \) be a continuous function and locally of bounded variation on some not necessarily bounded interval \( \left\lbrack {A, B}\right\rbrack \) . Then\n\n\[ \mathop{\sum }\limits_{{A \leq n \leq B}}^{\prime }f\left( n\right) = \mathop{\sum }\limits_{{m \in \mathbb{Z... | Proof. Let \( {f}_{1} \) be a piecewise continuous function locally of bounded variation, that tends to zero sufficiently rapidly (we will in fact have \( {f}_{1} \) with compact support, so this is no problem). Set \( g\left( x\right) = \mathop{\sum }\limits_{{n \in \mathbb{Z}}}{f}_{1}\left( {n + x}\right) \) . Then \... | Yes |
Corollary 2.2.17. Let \( f \) be a continuous function and locally of bounded variation on \( \mathbb{R} \). Then for all \( x \in \mathbb{R} \) we have\n\n\[ \mathop{\sum }\limits_{{n \in \mathbb{Z}}}f\left( {x + n}\right) = \mathop{\sum }\limits_{{m \in \mathbb{Z}}}\widehat{f}\left( m\right) \exp \left( {2i\pi mx}\ri... | Proof. Apply the proposition to \( \left\lbrack {A, B}\right\rbrack = \mathbb{R} \), and note that by an evident change of variable the Fourier transform of \( f\left( {x + t}\right) \) at \( y \) is \( \widehat{f}\left( y\right) {e}^{2i\pi yx} \). | Yes |
Lemma 2.2.18. Let \( p \) be an odd prime number, and let \( \chi \left( n\right) = \left( \frac{n}{p}\right) \) be the Legendre symbol. Then\n\n\[ \n\tau \left( \chi \right) = \mathop{\sum }\limits_{{x{\;\operatorname{mod}\;p}}}{\zeta }_{p}^{{x}^{2}} \n\] | Proof. This immediately follows from the trivial observation that the number of solutions modulo \( p \) to \( {x}^{2} \equiv n\left( {\;\operatorname{mod}\;p}\right) \) is equal to \( 1 + \chi \left( n\right) \) and the fact that \( \mathop{\sum }\limits_{{n{\;\operatorname{mod}\;p}}}\chi \left( n\right) = 0 \) . | Yes |
Theorem 2.2.19. Let \( p \) be an odd prime number, and let \( \chi \left( n\right) = \left( \frac{n}{p}\right) \) be the Legendre symbol. Then\n\n\[ \n\tau \left( \chi \right) = \left\{ \begin{array}{ll} {p}^{1/2} & \text{ if }p \equiv 1\left( {\;\operatorname{mod}\;4}\right) \\ {p}^{1/2}i & \text{ if }p \equiv 3\left... | Proof. By the above lemma, we have \( \tau \left( \chi \right) = \mathop{\sum }\limits_{{0 \leq x \leq p - 1}}\exp \left( {{2i\pi }{x}^{2}/p}\right) \) . We apply the Poisson summation formula proved above to \( \left\lbrack {A, B}\right\rbrack = \left\lbrack {0, p}\right\rbrack \) and \( f\left( x\right) = \exp \left(... | Yes |
Lemma 2.2.20. We have \( \tau \left( {\chi }_{D}\right) = {D}^{1/2} \) for \( D = - 4, D = - 8 \), and \( D = 8 \) . | Proof. Clear by direct computation. | No |
Lemma 2.2.21. Let \( {D}_{1} \) and \( {D}_{2} \) be two coprime fundamental discriminants. If \( \tau \left( {\chi }_{{D}_{1}}\right) = {D}_{1}^{1/2} \) and \( \tau \left( {\chi }_{{D}_{2}}\right) = {D}_{2}^{1/2} \), then \( \tau \left( {\chi }_{{D}_{1}{D}_{2}}\right) = {\left( {D}_{1}{D}_{2}\right) }^{1/2} \) . | Proof. First note the important fact that it is not true that \( {\left( {D}_{1}{D}_{2}\right) }^{1/2} = \) \( {D}_{1}^{1/2}{D}_{2}^{1/2} \) (example \( {D}_{1} = - 3,{D}_{2} = - 7 \) ).\n\nSince \( {D}_{1} \) and \( {D}_{2} \) are coprime, by the Chinese remainder theorem a residue modulo \( {D}_{1}{D}_{2} \) can be w... | Yes |
Lemma 2.2.23. Any fundamental discriminant \( D \) can be written in a unique way as a product of prime fundamental discriminants. | Proof. Since \( D \) is fundamental, no odd prime can divide \( D \) to a power larger than 1. Thus, we may write \( D = {2}^{u}\mathop{\prod }\limits_{{p \in S}}p \), where \( S \) is a finite set of odd primes. It follows that \( D = \varepsilon {2}^{u}\mathop{\prod }\limits_{{p \in S}}{\left( -1\right) }^{\left( {p ... | Yes |
Proposition 2.2.24. Let \( \chi \) be a real primitive character modulo \( m \), so that \( \chi \left( n\right) = \left( \frac{D}{n}\right) \) for \( D = \chi \left( {-1}\right) m \) a fundamental discriminant. Then\n\n\[ \n\tau \left( \chi \right) = \left\{ \begin{array}{ll} {m}^{1/2} & \text{ if }\chi \left( {-1}\ri... | Proof. By Theorem 2.2.15, we know that \( \chi = {\chi }_{D} \) with \( D = \chi \left( {-1}\right) m \) a fundamental discriminant. By Lemma 2.2.23, \( D \) is equal to a product of prime fundamental discriminants that are necessarily coprime. By Lemma 2.2.21, it is thus sufficient to prove the proposition for prime f... | No |
Proposition 2.3.3. Let \( \Lambda \) be a lattice in \( V \) and let \( {\left( {\mathbf{b}}_{j}\right) }_{1 \leq j \leq n} \) be a \( \mathbb{Z} \) -basis of \( \Lambda \). (1) The quantity \( \det \left( {{\mathbf{b}}_{1},\ldots ,{\mathbf{b}}_{n}}\right) \) is independent of the choice of the \( \mathbb{Z} \) -basis ... | Proof. (1). If \( {\mathbf{b}}_{j}^{\prime } \) is another \( \mathbb{Z} \) -basis of \( \Lambda \) the transition matrix from the \( {\mathbf{b}}_{i} \) to the \( {\mathbf{b}}_{j}^{\prime } \) is a matrix \( P \) with integral entries whose inverse also has integral entries, hence is such that \( \det \left( P\right) ... | Yes |
Corollary 2.3.4. Let \( {\mathbf{b}}_{1},\ldots ,{\mathbf{b}}_{n} \) belong to a lattice \( \Lambda \), and let \( \mathcal{B} \) be the matrix of the \( {\mathbf{b}}_{j} \) on some orthonormal matrix of \( V \) . The \( \left( {\mathbf{b}}_{i}\right) \) form a \( \mathbb{Z} \) -basis of \( \Lambda \) if and only if \(... | Proof. Clear. | No |
Proposition 2.3.5. Let \( {\left( {\mathbf{b}}_{j}\right) }_{1 \leq j \leq n} \) be an \( \mathbb{R} \) -basis of \( V \) . There exists a unique orthogonal (but not necessarily orthonormal) basis \( {\left( {\mathbf{b}}_{j}^{ * }\right) }_{1 \leq j \leq n} \) of \( V \) whose matrix on the \( {\mathbf{b}}_{i} \) is up... | Proof. The transition matrix is upper triangular with 1 on the diagonal if and only if its inverse is also of this form, hence if and only if \( {\mathbf{b}}_{i}^{ * } = {\mathbf{b}}_{i} - \) \( \mathop{\sum }\limits_{{1 \leq j < i}}{\mu }_{i, j}{\mathbf{b}}_{j}^{ * } \) for some \( {\mu }_{i, j} \in \mathbb{R} \) . Th... | Yes |
Corollary 2.3.7 (Hadamard’s inequality). Let \( \\left( {\\mathbf{b}}_{j}\\right) \) be an \( \\mathbb{R} \) -basis of \( V \) and let \( \\left( {\\mathbf{b}}_{j}^{ * }\\right) \) be the associated Gram-Schmidt basis of \( V \) . We have\n\n\[ \n\\det \\left( {{\\mathbf{b}}_{1},\\ldots ,{\\mathbf{b}}_{n}}\\right) = \\... | Proof. Since the transition matrix from the \( \\left( {\\mathbf{b}}_{j}\\right) \) to the \( \\left( {\\mathbf{b}}_{j}^{ * }\\right) \) has determinant 1, we have \( \\det \\left( {{\\mathbf{b}}_{1},\\ldots ,{\\mathbf{b}}_{n}}\\right) = \\det \\left( {{\\mathbf{b}}_{1}^{ * },\\ldots ,{\\mathbf{b}}_{n}^{ * }}\\right) \... | Yes |
Lemma 2.3.8. Let \( \left( {{\mathbf{b}}_{1},\ldots ,{\mathbf{b}}_{n}}\right) \) be an \( \mathbb{R} \) -basis of \( V \), let \( W = {\mathbf{b}}_{1}^{ \bot } \) be the orthogonal supplement of \( {\mathbf{b}}_{1} \), and let \( {\mathbf{b}}_{2}^{\prime },\ldots ,{\mathbf{b}}_{n}^{\prime } \) be the orthogonal project... | Proof. Let \( \left( {{e}_{2},\ldots ,{e}_{n}}\right) \) be an orthonormal basis of \( W \), so that if we set \( {e}_{1} = {\mathbf{b}}_{1}/\begin{Vmatrix}{\mathbf{b}}_{1}\end{Vmatrix},\left( {{e}_{1},\ldots ,{e}_{n}}\right) \) is an orthonormal basis of \( V \) . For \( j \geq 2 \) we thus have \( {\mathbf{b}}_{j} = ... | Yes |
Corollary 2.3.9. Let \( \Lambda \) be a lattice in \( V \), let \( {\mathbf{b}}_{1} \) be an element of a \( \mathbb{Z} \) -basis of \( \Lambda \), let \( W = {\mathbf{b}}_{1}^{ \bot } \) be its orthogonal supplement, and let \( {\Lambda }^{\prime } \) be the projection of \( \Lambda \) on \( W \) . Then \( {\Lambda }^... | Proof. Applying the above lemma to a \( \mathbb{Z} \) -basis \( \left( {{\mathbf{b}}_{1},\ldots ,{\mathbf{b}}_{n}}\right) \) of \( \Lambda \), it is clear that \( \left( {{\mathbf{b}}_{2}^{\prime },\ldots ,{\mathbf{b}}_{n}^{\prime }}\right) \) satisfy conditions (1) and (3) of Proposition 2.3.1; hence \( {\Lambda }^{\p... | Yes |
Lemma 2.3.11. Keep the notation of the above corollary, and assume that \( {\mathbf{b}}_{1} \) is a nonzero vector of \( \Lambda \) with minimal norm. Then every \( {x}^{\prime } \in {\Lambda }^{\prime } \) is the orthogonal projection of some \( x \in \Lambda \) such that \( \parallel x{\parallel }^{2} \leq \left( {4/... | Proof. We may of course assume that \( {x}^{\prime } \neq 0 \) . Let \( {x}_{0} \) be any element of \( \Lambda \) that projects on \( {x}^{\prime } \), so that \( {x}_{0} = {x}^{\prime } - \alpha {\mathbf{b}}_{1} \) for some \( \alpha \in \mathbb{R} \) . The elements of \( \Lambda \) that project on \( {x}^{\prime } \... | Yes |
Theorem 2.3.12 (Hermite’s inequality). Let \( \Lambda \) be a lattice in \( V \) . There exists a \( \mathbb{Z} \) -basis \( \left( {{\mathbf{b}}_{1},\ldots ,{\mathbf{b}}_{n}}\right) \) of \( \Lambda \) such that\n\n\[ \det \left( \Lambda \right) \leq \mathop{\prod }\limits_{{j = 1}}^{n}\begin{Vmatrix}{\mathbf{b}}_{j}\... | Proof. The first inequality is simply Hadamard's inequality (Corollary 2.3.7). We prove the second one by induction on \( n \), the case \( n = 1 \) being trivial. Let \( n \geq 2 \), assume the result true up to \( n - 1 \), let \( {\mathbf{b}}_{1} \) be a nonzero vector of \( \Lambda \) with minimal norm, and keep th... | Yes |
Corollary 2.3.13 (Fermat). Every prime \( p \equiv 1\left( {\;\operatorname{mod}\;4}\right) \) is the sum of two squares of integers. | Proof. See Exercise 41. | No |
Proposition 2.3.15. Let \( \left( {\mathbf{b}}_{j}\right) \) be a \( \gamma \) -LLL-reduced basis of \( \Lambda \), and let \( \left( {\mathbf{b}}_{j}^{ * }\right) \) be the corresponding Gram-Schmidt basis of \( {\mathbb{R}}^{n} \) . (1) For \( 1 \leq j \leq i \leq n \) we have \( {\begin{Vmatrix}{\mathbf{b}}_{j}\end{... | Proof. (1). Since \( \left| {\mu }_{i, i - 1}\right| \leq 1/2 \) we have \( {\begin{Vmatrix}{\mathbf{b}}_{i}^{ * }\end{Vmatrix}}^{2} \geq {\begin{Vmatrix}{\mathbf{b}}_{i - 1}^{ * }\end{Vmatrix}}^{2}/\gamma \) ; hence by induction, for \( j \leq i \) we have \( {\begin{Vmatrix}{\mathbf{b}}_{j}^{ * }\end{Vmatrix}}^{2} \l... | Yes |
Corollary 2.3.16. Let \( \left( {\mathbf{b}}_{i}\right) \) be a \( \gamma \) -LLL-reduced basis of \( \Lambda \) and let \( \left( {\mathbf{b}}_{i}^{ * }\right) \) be the corresponding Gram-Schmidt basis. Set\n\n\[ \n{c}_{1} = \mathop{\max }\limits_{{1 \leq i \leq n}}\frac{\begin{Vmatrix}{\mathbf{b}}_{1}\end{Vmatrix}}{... | Proof. Since \( {\begin{Vmatrix}{\mathbf{b}}_{1}^{ * }\end{Vmatrix}}^{2} \leq {\begin{Vmatrix}{\mathbf{b}}_{1}\end{Vmatrix}}^{2} \) we have \( {c}_{1} \geq 1 \), while by (1) of the proposition we have \( {\begin{Vmatrix}{\mathbf{b}}_{1}\end{Vmatrix}}^{2} \leq {\gamma }^{i - 1}{\begin{Vmatrix}{\mathbf{b}}_{i}^{ * }\end... | Yes |
Corollary 2.3.17. Let \( \left( {\mathbf{b}}_{i}\right) \) be a \( \gamma \) -LLL-reduced basis of \( \Lambda \), let \( \mathbf{y} \notin \Lambda \), let \( Y = \left( {y}_{i}\right) \) be the vector of coordinates of \( \mathbf{y} \) on the basis of the \( \left( {\mathbf{b}}_{i}\right) \), and let \( {i}_{0} \) be t... | Proof. We use essentially the same proof as the preceding corollary. We write as above\n\n\[ \mathbf{x} = \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}{\mathbf{b}}_{i} = \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}^{ * }{\mathbf{b}}_{i}^{ * } \]\n\n\[ \mathbf{y} = \mathop{\sum }\limits_{{i = 1}}^{n}{y}_{i}{\mathbf{b}}_{i} =... | Yes |
Theorem 2.3.19. There exists a polynomial-time algorithm that, given a basis of a lattice \( \Lambda \) outputs an LLL-reduced basis of \( \Lambda \) . Furthermore, if \( \Lambda \) is a sublattice of \( {\mathbb{Z}}^{n} \) (or more generally if the Gram matrix of a basis of \( \Lambda \) has integral entries) all the ... | Since we always assume that the reader has a number theory package at his disposal, we mention that in GP the commands are qf111(B) for the general LLL algorithm on a matrix \( B \), and qf111 (B,1) for the integral version, which is the one which must be used in the context of Diophantine applications. The output \( H... | No |
Proposition 2.3.20. Keep the above notation, and in particular assume that the \( {\alpha }_{i} \) are all real. Let \( {X}_{1},\ldots ,{X}_{n} \) be strictly positive integers, set \( Q = \) \( \mathop{\sum }\limits_{{1 \leq i \leq n - 1}}{X}_{i}^{2}, T = \left( {1 + \mathop{\sum }\limits_{{1 \leq i \leq n}}{X}_{i}}\r... | Proof. If we set\n\n\[ S = {\alpha }_{0} + \mathop{\sum }\limits_{{1 \leq i \leq n}}{x}_{i}{\alpha }_{i}\;\text{ and }\;K = \left\lfloor {C{\alpha }_{0}}\right\rceil + \mathop{\sum }\limits_{{1 \leq i \leq n}}{x}_{i}\left\lfloor {C{\alpha }_{i}}\right\rceil \]\n\nthen by definition \( \left| {K - {CS}}\right| \leq 1/2 ... | Yes |
Theorem 2.3.21 (Blichfeldt). Let \( S \) be a (measurable) subset of \( {\mathbb{R}}^{n} \) with volume \( \operatorname{Vol}\left( S\right) \), and let \( \Lambda \) be a lattice of \( {\mathbb{R}}^{n} \). If \( \operatorname{Vol}\left( S\right) > \det \left( \Lambda \right) \) there exist distinct elements \( a \) an... | Proof. Let \( {\mathbf{b}}_{1},\ldots ,{\mathbf{b}}_{n} \) be a \( \mathbb{Z} \)-basis of \( \Lambda \), let as above \( U = \{ x = \left. {\mathop{\sum }\limits_{{1 \leq j \leq n}}{x}_{j}{\mathbf{b}}_{j}/0 \leq {x}_{j} < 1}\right\} \) be a fundamental parallelotope of \( \Lambda \), and let \( \chi \left( x\right) \) ... | Yes |
Theorem 2.3.22 (Minkowski). Let \( C \subset {\mathbb{R}}^{n} \) be symmetric and convex, let \( \Lambda \) be a lattice in \( {\mathbb{R}}^{n} \), and assume that \( \operatorname{Vol}\left( C\right) > {2}^{n}\det \left( \Lambda \right) \) . Then there exists \( c \neq 0 \) such that \( c \in \Lambda \cap C \) . | Proof. Let \( S = C/2 = \{ x/2, x \in C\} \) be the homothetic of \( C \) by a factor \( 1/2 \), so that \( \operatorname{Vol}\left( S\right) > \det \left( \Lambda \right) \) . By Blichfeldt’s theorem there exist \( a \) and \( b \) in \( S \) such that \( c = a - b \in \Lambda \) with \( c \neq 0 \) . Thus \( {2a} \) ... | Yes |
Corollary 2.3.23. With the same assumptions, if in addition \( C \) is compact, the conclusion of the theorem still holds if we only have \( \operatorname{Vol}\left( C\right) \geq {2}^{n}\det \left( \Lambda \right) \) . | Proof. Applying Minkowski’s theorem to the homothetic set \( \left( {1 + \varepsilon }\right) C \) for any \( \varepsilon > 0 \), we see that there exists \( {c}_{\varepsilon } \in \Lambda \smallsetminus \{ 0\} \) such that \( {\left( 1 + \varepsilon \right) }^{-1}{c}_{\varepsilon } \in C \) . By compactness, the \( {\... | Yes |
Corollary 2.3.24. For \( 1 \leq j \leq n \), let \( {L}_{j}\left( y\right) = \mathop{\sum }\limits_{{1 \leq i \leq n}}{a}_{j, i}{y}_{i} \) be a linear form in the \( n \) variables \( {y}_{i} \) with real coefficients, and set \( \Delta = \left| {\det \left( {a}_{j, i}\right) }\right| \) . Let \( C \) be symmetric and ... | Proof. Set \( D = \left\{ {y \in {\mathbb{R}}^{n}/\left( {{L}_{1}\left( y\right) ,\ldots ,{L}_{n}\left( y\right) }\right) \in C}\right\} \) . Clearly \( D \) is symmetric and convex \( \left( {{tL}\left( y\right) + \left( {1 - t}\right) L\left( z\right) = L\left( {{ty} + \left( {1 - t}\right) z}\right) \text{if}L\text{... | Yes |
Corollary 2.3.25 (Minkowski). If \( \Lambda \) is a lattice in \( {\mathbb{R}}^{n} \) we have\n\n\[ \min \left( \Lambda \right) \leq \frac{2}{{\pi }^{1/2}}\Gamma {\left( \frac{n}{2} + 1\right) }^{1/n}\det {\left( \Lambda \right) }^{1/n}, \]\n\nwhere \( \Gamma \left( x\right) \) is the gamma function (see Chapter 9). | Proof. We choose for \( C = {C}_{\lambda } \) the closed ball centered at the origin with radius \( \lambda \), where \( \lambda \) will be chosen presently. It is clear that \( {C}_{\lambda } \) is convex, symmetric, and compact; hence if \( \operatorname{Vol}\left( {C}_{\lambda }\right) \geq {2}^{n}\det \left( \Lambd... | Yes |
Any finite subgroup of the multiplicative group of a commutative field \( K \) is cyclic. In particular, the multiplicative group of a finite field is cyclic; in other words, if \( K \) is a finite field with \( {p}^{n} \) elements then \[ \left( {{K}^{ * }, \times }\right) \simeq \left( {\mathbb{Z}/\left( {{p}^{n} - 1... | Proof. Let \( G \) be such a finite subgroup, say of order \( n \) . For every \( d \mid n \) , let \( \rho \left( d\right) \) be the number of \( x \in G \) of order exactly equal to \( d \) in \( G \) . We clearly have \( n = \mathop{\sum }\limits_{{d \mid n}}\rho \left( d\right) \) . On the other hand, since in a co... | Yes |
Corollary 2.4.4. Let \( y \in {\mathbb{F}}_{q} \) and \( m \in {\mathbb{Z}}_{ \geq 1} \) . (1) The number of solutions in \( {\mathbb{F}}_{q} \) of the equation \( {x}^{m} = y \) is equal to the number of solutions of \( {x}^{d} = y \), where \( d = \gcd \left( {m, q - 1}\right) \) . | Proof. If \( y = 0 \) there is the unique solution \( x = 0 \), so we may assume that \( y \neq 0 \) . Since the group \( {\mathbb{F}}_{q}^{ * } \) is cyclic, the image of the map \( x \mapsto {x}^{m} \) is \( {\mathbb{F}}_{q}^{*d} \) , the subgroup of \( d \) th powers, and for each \( y \in {\mathbb{F}}_{q}^{*d} \) i... | Yes |
For any integer \( n \geq 1 \) there exists a finite subfield of \( \overline{{\mathbb{F}}_{p}} \) with \( q = {p}^{n} \) elements. This subfield is unique and is equal to the set of roots in \( \overline{{\mathbb{F}}_{p}} \) of the equation \( {X}^{q} - X = 0 \) . Up to isomorphism, there exists a unique finite field ... | Assume first that a subfield \( F \) of \( \overline{{\mathbb{F}}_{p}} \) with \( q \) elements exists. Since \( \left| {F}^{ * }\right| = q - 1 \), any element \( x \in {F}^{ * } \) satisfies the equation \( {x}^{q - 1} = 1 \), hence any element \( x \in F \) satisfies the equation \( {x}^{q} - x = 0 \) . Conversely, ... | Yes |
Proposition 2.4.7. If \( n \) and \( m \) are in \( {\mathbb{Z}}_{ \geq 1} \) then\n\n\[{\mathbb{F}}_{{p}^{n}} \subset {\mathbb{F}}_{{p}^{m}} \Leftrightarrow n \mid m.\]\n\nIn particular, \( {\mathbb{F}}_{{p}^{n}} \cap {\mathbb{F}}_{{p}^{m}} = {\mathbb{F}}_{{p}^{\gcd \left( {n, m}\right) }} \) and \( {\mathbb{F}}_{{p}^... | Proof. Left as an easy exercise to the reader. | No |
Theorem 2.4.8. Let \( \mathbb{E}/\mathbb{F} \) be an extension of finite fields. Then \( \mathbb{E}/\mathbb{F} \) is a Galois (i.e., normal and separable) extension and the Galois group \( \operatorname{Gal}\left( {\mathbb{E}/\mathbb{F}}\right) \) of \( \mathbb{F} \) -automorphisms of \( \mathbb{E} \) is the cyclic gro... | Proof. Up to isomorphism, we may assume that we are in a fixed algebraic closure \( \overline{{\mathbb{F}}_{p}} \) of \( {\mathbb{F}}_{p} \), and that \( \mathbb{F} = {\mathbb{F}}_{q} \) (with \( q = {p}^{f} \) for some \( f \) ) and \( \mathbb{E} = {\mathbb{F}}_{{q}^{s}} \) for \( s = \left\lbrack {\mathbb{E} : \mathb... | Yes |
Corollary 2.4.9. Let \( \mathbb{E}/\mathbb{F} \) be an extension of finite fields, \( q = \left| \mathbb{F}\right| \), and \( s = \) \( \left\lbrack {\mathbb{E} : \mathbb{F}}\right\rbrack \) . The trace and norm from \( \mathbb{E} \) to \( \mathbb{F} \) are given by the formulas | \[ {\operatorname{Tr}}_{\mathbb{E}/\mathbb{F}}\left( x\right) = \mathop{\sum }\limits_{{0 \leq i < s}}{x}^{{q}^{i}} \] \[ {\mathcal{N}}_{\mathbb{E}/\mathbb{F}}\left( x\right) = {x}^{\left( {{q}^{s} - 1}\right) /\left( {q - 1}\right) }.\] | Yes |
Corollary 2.4.10. For any \( q = {p}^{f} \), the subfield \( {\mathbb{F}}_{q} \) of \( \overline{{\mathbb{F}}_{p}} \) is the fixed field of \( {\sigma }_{p}^{f} \), and | \[ \operatorname{Gal}\left( {{\mathbb{F}}_{q}/{\mathbb{F}}_{p}}\right) = {\sigma }_{p}^{\mathbb{Z}}/{\sigma }_{p}^{f\mathbb{Z}} \simeq \mathbb{Z}/f\mathbb{Z}. \] | No |
Proposition 2.4.11. Let \( \mathbb{E}/\mathbb{F} \) be an extension of finite fields with \( \left| \mathbb{F}\right| = q \) .\n\n(1) The trace map \( {\operatorname{Tr}}_{\mathbb{E}/\mathbb{F}} \) is a surjective homomorphism from \( \mathbb{E} \) to \( \mathbb{F} \) . | Proof. (1). By Corollary 2.4.9 we have\n\n\[ \n{\operatorname{Tr}}_{\mathbb{E}/\mathbb{F}}\left( x\right) = \mathop{\sum }\limits_{{0 \leq i < s}}{x}^{{q}^{i}} \n\] \n\nThe right-hand side is a polynomial of degree \( {q}^{s - 1} \), hence has at most \( {q}^{s - 1} < \) \( {q}^{s} = \left| \mathbb{E}\right| \) roots i... | Yes |
Proposition 2.4.12. Let \( \mathbb{E}/\mathbb{F} \) be an extension of finite fields. The norm map \( {\mathcal{N}}_{\mathbb{E}/\mathbb{F}} \) is a surjective homomorphism from \( {\mathbb{E}}^{ * } \) to \( {\mathbb{F}}^{ * } \) . | Proof. Let \( g \) be a generator of the cyclic group \( {\mathbb{E}}^{ * } \) . The subgroup \( {\mathbb{F}}^{ * } \) is the unique subgroup of cardinality \( q - 1 \) ; hence it is the group generated by \( {g}^{\left( {{q}^{s} - 1}\right) /\left( {q - 1}\right) } \) . Thus if \( a \in {\mathbb{F}}^{ * } \) we can wr... | Yes |
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