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Proposition 17. If \( {\psi }_{1} \) and \( {\psi }_{2} \) are characters, then so is their product \( {\psi }_{1}{\psi }_{2} \) . | Proof: Let \( {V}_{1} \) and \( {V}_{2} \) be \( \mathbb{C}G \) -modules affording characters \( {\psi }_{1} \) and \( {\psi }_{2} \) and define \( W = {V}_{1}{ \otimes }_{\mathbb{C}}{V}_{2} \) . Since each \( g \in G \) acts as a linear transformation on \( {V}_{1} \) and \( {V}_{2} \), the action of \( g \) on simple... | No |
Proposition 2. Let \( \alpha \in \mathbb{C} \) .\n\n(1) The following are equivalent:\n\n(i) \( \alpha \) is an algebraic integer,\n\n(ii) \( \alpha \) is algebraic over \( \overline{\mathbb{Q}} \) and the minimal polynomial of \( \alpha \) over \( \mathbb{Q} \) has integer coefficients, and\n\n(iii) \( \mathbb{Z}\left... | Proof: These are established in Section 15.3. (The portion of Section 15.3 consisting of integral extensions and properties of algebraic integers may be read independently from the rest of Chapter 15.) | No |
Corollary 3. For every character \( \psi \) of the finite group \( G,\psi \left( x\right) \) is an algebraic integer for all \( x \in G \) . | Proof: By Proposition 14 in Section 18.3, \( \psi \left( x\right) \) is a sum of roots of 1 . Each root of 1 is an algebraic integer, so the result follows immediately from Proposition 2(2). | Yes |
Define the complex valued function \( {\omega }_{i} \) on \( \left\{ {{\mathcal{K}}_{1},\ldots ,{\mathcal{K}}_{r}}\right\} \) for each \( i \) by\n\n\[ \n{\omega }_{i}\left( {\mathcal{K}}_{j}\right) = \frac{\left| {\mathcal{K}}_{j}\right| {\chi }_{i}\left( g\right) }{{\chi }_{i}\left( 1\right) }\n\]\n\nwhere \( g \) is... | We first prove that if \( I \) is the identity matrix, then\n\n\[ \n\mathop{\sum }\limits_{{g \in {\mathcal{K}}_{j}}}{\varphi }_{i}\left( g\right) = {\omega }_{i}\left( {\mathcal{K}}_{j}\right) I\n\]\n\n(19.1)\n\n\n\nTo see this let \( X \) be the left hand side of (1). As we saw in Section 18.2, each \( x \in G \) act... | Yes |
Corollary 5. The degree of each complex irreducible representation of a finite group \( G \) divides the order of \( G \), i.e., \( {\chi }_{i}\left( 1\right) \left| \right| G \mid \) for \( i = 1,2,\ldots, r \) . | Proof: Under the notation of Proposition 4 and with \( {g}_{j} \in {\mathcal{K}}_{j} \) we have\n\n\[ \frac{\left| G\right| }{{\chi }_{i}\left( 1\right) } = \frac{\left| G\right| }{{\chi }_{i}\left( 1\right) }\left( {{\chi }_{i},{\chi }_{i}}\right) \]\n\n\[ = \mathop{\sum }\limits_{{j = 1}}^{r}\frac{\left| {\mathcal{K}... | Yes |
Lemma 6. If \( G \) is any group that has a conjugacy class \( \mathcal{K} \) and an irreducible matrix representation \( \varphi \) with character \( \chi \) such that \( \left( {\left| \mathcal{K}\right| ,\chi \left( 1\right) }\right) = 1 \), then for \( g \in \mathcal{K} \) either \( \chi \left( g\right) = 0 \) or \... | Proof: By hypothesis there exist \( s, t \in \mathbb{Z} \) such that \( s\left| \mathcal{K}\right| + {t\chi }\left( 1\right) = 1 \) . Thus\n\n\[ s\left| \mathcal{K}\right| \chi \left( g\right) + {t\chi }\left( 1\right) \chi \left( g\right) = \chi \left( g\right) .\n\]\n\nDivide both sides of this by \( \chi \left( 1\ri... | Yes |
Lemma 7. If \( \left| \mathcal{K}\right| \) is a power of a prime for some nonidentity conjugacy class \( \mathcal{K} \) of \( G \) , then \( G \) is not a non-abelian simple group. | Proof: Suppose to the contrary that \( G \) is a non-abelian simple group and let \( \left| \mathcal{K}\right| = {p}^{c} \) . Let \( g \in \mathcal{K} \) . If \( c = 0 \) then \( g \in Z\left( G\right) \), contrary to a non-abelian simple group having a trivial center. As above, let \( {\chi }_{1},\ldots ,{\chi }_{r} \... | Yes |
Lemma 9. If \( G \) is solvable of order \( > 1 \), then there exists \( P \trianglelefteq G \) with \( P \) a nontrivial \( p \) -group for some prime \( p \) . | Proof: This is a special case of the exercise on minimal normal subgroups of solvable groups at the end of Section 6.1. One can see this easily by letting \( P \) be a nontrivial Sylow subgroup of the last nontrivial term, \( {G}^{\left( n - 1\right) } \), in the derived series of \( G \) (where \( G \) has solvable le... | No |
Lemma 10. Let \( G \) be a group of order \( {p}_{1}^{{\alpha }_{1}}{p}_{2}^{{\alpha }_{2}}\cdots {p}_{t}^{{\alpha }_{t}} \) where \( {p}_{1},\ldots ,{p}_{t} \) are distinct primes. Suppose there are subgroups \( H \) and \( \bar{K} \) of \( G \) such that for each \( i \in \{ 1,\ldots, t\} \) , either \( {p}_{i}^{{\al... | Proof: Fix some \( i \in \{ 1,\ldots, t\} \) and suppose first that \( {p}_{i}^{{\alpha }_{i}} \) divides the order of \( H \) . Since \( {HK} \) is a disjoint union of right cosets of \( H \) and each of these right cosets has order equal to \( \left| H\right| \), it follows that \( {p}_{i}^{{\bar{\alpha }}_{i}} \) di... | Yes |
Theorem 11. Let \( H \) be a subgroup of the finite group \( G \) and let \( {g}_{1},\ldots ,{g}_{m} \) be representatives for the distinct left cosets of \( H \) in \( G \) . Let \( V \) be an \( {FH} \) -module affording the matrix representation \( \varphi \) of \( H \) of degree \( n \) . The \( {FG} \) -module \( ... | Proof: First note that \( {FG} \) is a free right \( {FH} \) -module:\n\n\[ {FG} = {g}_{1}{FH} \oplus {g}_{2}{FH} \oplus \cdots \oplus {g}_{m}{FH}. \]\n\nSince tensor products commute with direct sums (Theorem 17, Section 10.4), as abelian groups we have\n\n\[ W = {FG}{ \otimes }_{FH}V \cong \left( {{g}_{1} \otimes V}\... | Yes |
In the notation of Theorem 11, (1) if \( \psi \) is the character afforded by \( V \) then the induced character is given by \[ {\operatorname{Ind}}_{H}^{G}\left( \psi \right) \left( g\right) = \mathop{\sum }\limits_{{i = 1}}^{m}\psi \left( {{g}_{i}^{-1}g{g}_{i}}\right) \] where \( \psi \left( {{g}_{i}^{-1}g{g}_{i}}\ri... | Proof: From the matrix of \( g \) computed above, the blocks \( \varphi \left( {{g}_{i}^{-1}g{g}_{i}}\right) \) down the diagonal of \( \Phi \left( g\right) \) are zero except when \( {g}_{i}^{-1}g{g}_{i} \in H \). Thus the trace of the block matrix \( \Phi \left( g\right) \) is the sum of the traces of the matrices \(... | Yes |
Proposition 13. Let \( G \) be a Frobenius group of order \( {q}^{a}p \), where \( p \) and \( q \) are distinct primes, such that the Frobenius kernel \( Q \) is an elementary abelian \( q \) -group of order \( {q}^{a} \) and the cyclic group \( G/Q \) acts irreducibly by conjugation on \( Q \) . Then the following ho... | Proof: Note that \( {QP} \) equals \( G \) by order consideration. By definition of a Frobenius group and because \( Q \) is abelian, \( {C}_{G}\left( h\right) = Q \) for every nonidentity element \( h \) of \( Q \) . If \( x \) were an element of order \( {pq} \), then \( {x}^{p} \) would be an element of order \( q \... | Yes |
Proposition 14. Let \( G \) be a group, let \( H \) be a subgroup of \( G \) and let \( \psi \) and \( {\psi }^{\prime } \) be characters of \( H \) .\n\n(1) (Induction of characters is additive) \( {\operatorname{Ind}}_{H}^{G}\left( {\psi + {\psi }^{\prime }}\right) = {\operatorname{Ind}}_{H}^{G}\left( \psi \right) + ... | It follows from part (1) of Proposition 14 that if \( \mathop{\sum }\limits_{{i = 1}}^{s}{n}_{i}{\psi }_{i} \) is any integral linear combination of characters of \( H \) with \( {n}_{i} \geq 0 \) for all \( i \) then\n\n\[ {\operatorname{Ind}}_{H}^{G}\left( {\mathop{\sum }\limits_{{i = 1}}^{s}{n}_{i}{\psi }_{i}}\right... | Yes |
For any \( i \in \{ 1,2,3,4\} \) let \( q = {q}_{i} \), let \( Q = {Q}_{i} \), let \( N = {N}_{i} \) and let \( p = \left| {N : Q}\right| \) . Let \( {\psi }_{1},\ldots ,{\psi }_{4} \) be any irreducible characters of \( N \) of degree \( p \) (not necessarily distinct) and let \( \alpha = {\psi }_{1} - {\psi }_{2} \) ... | Proof: By Proposition 13, there are nonprincipal characters \( {\lambda }_{1},\ldots ,{\lambda }_{4} \) of \( Q \) of degree 1 such that \( {\psi }_{j} = {\operatorname{Ind}}_{Q}^{N}\left( {\lambda }_{j}\right) \) for \( j = 1,\ldots ,4 \) . By Corollary 12 therefore, each \( {\psi }_{j} \) vanishes on \( N - Q \), hen... | Yes |
For any \( i \in \{ 1,2,3,4\} \) let \( q = {q}_{i} \), let \( Q = {Q}_{i} \), let \( N = {N}_{i} \) and let \( p = \left| {N : Q}\right| \) . Let \( {\psi }_{1},\ldots ,{\psi }_{k} \) be the distinct irreducible characters of \( N \) of degree \( p \) . Then there are distinct irreducible characters \( {\chi }_{1},\ld... | Proof: Let \( {\alpha }_{j} = {\psi }_{1} - {\psi }_{j} \) for \( j = 2,3,\ldots, k \) so \( {\alpha }_{j} \) satisfies the hypothesis of Lemma 15. Since \( {\psi }_{1} \neq {\psi }_{j} \), by Lemma 15\n\n\[ 2 = \left| \right| {\alpha }_{j}{\left| \right| }^{2} = {\left( {\alpha }_{j},{\alpha }_{j}\right) }_{N} = {\lef... | Yes |
Lemma 17. The exceptional characters associated to \( {Q}_{i} \) are all distinct from the exceptional characters associated to \( {Q}_{j} \) for \( i \) and \( j \) distinct elements of \( \{ 1,2,3,4\} \) . | Proof: Let \( \chi \) be an exceptional character associated to \( {Q}_{i} \) and let \( \theta \) be an exceptional character associated to \( {Q}_{j} \) . By construction, there are distinct irreducible characters \( \psi \) and \( {\psi }^{\prime } \) of \( {Q}_{i} \) such that \( {\psi }^{ * } - {\psi }^{\prime * }... | Yes |
Proposition 1. Let \( I \) be a nonempty countable set and for each \( i \in I \) let \( {A}_{i} \) be a set. The cardinality of the Cartesian product is the product of the cardinalities of the sets \( {A}_{i} \), i.e., \[ \left| {\mathop{\prod }\limits_{{i \in I}}{A}_{i}}\right| = \mathop{\prod }\limits_{{i \in I}}\le... | Proof: In order to count the number of choice functions note that each \( i \in I \) may be mapped to any of the \( \left| {A}_{i}\right| \) elements of \( {A}_{i} \) and for \( i \neq j \) the values of choice functions at \( i \) and \( j \) may be chosen completely independently. Thus the number of choice functions ... | Yes |
Theorem 2. Assuming the usual (Zermelo-Fraenkel) axioms of set theory, the following are equivalent: (1) Zorn's Lemma (2) the Axiom of Choice (3) the Well Ordering Principle. | Proof: This follows from elementary set theory. We refer the reader to Real and Abstract Analysis by Hewitt and Stromberg, Springer-Verlag, 1965, Section 3 for these equivalences and some others. | No |
Lemma 1 For all \( n \in \mathbb{N} \), we have\n\n\[ \n{S}_{n}f = \mathop{\sum }\limits_{{k = 0}}^{\infty }{T}^{{I}_{n\left( k\right) }^{k}}f = {\psi }_{n}\mathop{\sum }\limits_{{k = 0}}^{\infty }\mathop{\sum }\limits_{{l = 0}}^{{{n}_{k} - 1}}{\bar{r}}_{k}^{{n}_{k} - l}{E}_{k}\left( {{d}_{k + 1}\left( {f{\bar{\psi }}_... | Proof We sketch the proof, only. It is proved in [66] that\n\n\[ \n{T}^{{I}_{n\left( k\right) }^{k}}f = \mathop{\sum }\limits_{{j \in \lbrack n\left( {k + 1}\right), n\left( k\right) )}}\widehat{f}\left( j\right) {\psi }_{j}\n\]\n\n\[ \n= {\psi }_{n}\mathop{\sum }\limits_{{l = 0}}^{{{n}_{k} - 1}}{\bar{r}}_{k}^{{n}_{k} ... | No |
Lemma 2 For all \( k, n \in \mathbb{N} \), we have\n\n\[ \left| {{T}^{{I}_{n\left( k\right) }^{k}}f}\right| \leq R{E}_{k}\left( \left| {{s}_{{I}_{n\left( {k + 1}\right) }^{k + 1}}f - {s}_{{I}_{n\left( k\right) }^{k}}f}\right| \right) ,\]\n\nwhere \( R \mathrel{\text{:=}} \max \left( {{m}_{n}, n \in \mathbb{N}}\right) \... | Proof Equalities (9) and (6) imply\n\n\[ \left| {{T}^{{I}_{n\left( k\right) }^{k}}f}\right| \leq {m}_{k}{E}_{k}\left( \left| {{d}_{k + 1}\left( {f{\bar{\psi }}_{n}}\right) }\right| \right) \]\n\n(10)\n\n\[ \leq R{E}_{k}\left( \left| {{\psi }_{n}{E}_{k + 1}\left( {f{\overline{\psi }}_{n}}\right) - {\psi }_{n}{E}_{k}\lef... | Yes |
Lemma 3 For all \( n \in \mathbb{N},{\left( {\bar{\psi }}_{n}{T}^{{I}_{n\left( k\right) }^{k}}f\right) }_{k \in \mathbb{N}} \) is a martingale difference sequence with respect to \( {\left( {\mathcal{F}}_{k + 1}\right) }_{k \in \mathbb{N}} \) . | Proof First, \( {\bar{\psi }}_{n}{T}^{{I}_{n\left( k\right) }^{k}}f \) is \( {\mathcal{F}}_{k + 1} \) measurable because of (9) and the fact that \( {r}_{k} \) is \( {\mathcal{F}}_{k + 1} \) measurable. Since \( {E}_{k}\left( {r}_{k}^{i}\right) = 0 \) for \( i = 1,\ldots ,{m}_{n} - 1 \), we can see that \[ {E}_{k}\left... | Yes |
Theorem 2 Let \( f \in {L}_{p}\left( {G}_{m}\right) \), where \( 1 < p < \infty \) . Then\n\n\[{\begin{Vmatrix}{S}^{ * }f\end{Vmatrix}}_{p} \leq {c}_{p}\parallel f{\parallel }_{p}\]\n\nwhere\n\n\[{S}^{ * }f \mathrel{\text{:=}} \mathop{\sup }\limits_{{n \in \mathbb{N}}}\left| {{S}_{n}f}\right|\] | Proof It is easy to see that Lemma 1 implies \( {S}^{ * }f \leq {T}^{ * }f \) . It follows from Lemmas 6 and 7 that\n\n\[ \mathop{\sup }\limits_{{y > 0}}{y}^{p}\mu \left( {{S}^{ * }f > y}\right) \leq {C}_{p}\parallel f{\parallel }_{p} \]\n\nfor \( 1 < p < \infty \) . Now the proof of the theorem follows by the Marcinki... | Yes |
Theorem 4 Let \( f \in {L}_{p}\left( {G}_{m}\right) \), where \( p > 1 \) . Then\n\n\[ \n{S}_{n}f \rightarrow f,\;\text{ a.e., as }n \rightarrow \infty .\n\] | The proof follows directly by using Theorem 2 and the fact that the Vilinkin polynomials are dense in \( {L}_{p} \) . | Yes |
Lemma 8 If \( E \) is a set of divergence for \( {L}_{1}\left( {G}_{m}\right) \), then there is a function \( f \in {L}_{1}\left( {G}_{m}\right) \) such that \( {S}^{ * }f = \infty \) on \( E \) . | Proof We claim that given any \( g \in {L}_{1}\left( {G}_{m}\right) \), there is an unbounded monotone increasing sequence \( \lambda = \left( {{\lambda }_{j}, j \in \mathbb{N}}\right) \) of positive real numbers and a function \( f \in {L}_{1}\left( {G}_{m}\right) \) such that\n\n\[ \widehat{f}\left( j\right) = {\lamb... | Yes |
Corollary 1 If \( {E}_{1},{E}_{2},\ldots \) are sets of divergence for \( {L}_{1}\left( {G}_{m}\right) \), then\n\n\[ E \mathrel{\text{:=}} { \cup }_{n = 1}^{\infty }{E}_{n} \]\n\nis also a set of divergence for \( {L}_{1}\left( {G}_{m}\right) \) . | Proof Apply Lemma 9 to choose Vilenkin polynomials \( {P}_{1}^{\left( n\right) },{P}_{2}^{\left( n\right) },\ldots \) such that\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{\infty }{\begin{Vmatrix}{P}_{j}^{\left( n\right) }\end{Vmatrix}}_{1} < \infty \]\n\nand\n\n\[ \mathop{\sup }\limits_{{j \in {\mathbb{N}}_{ + }}}\left( {{... | Yes |
Theorem 4.6 (Stokes’ Theorem for holomorphic differentials). Suppose \( C \) is a Riemann surface with \( \Omega \subset C \) an open set, \( \bar{\Omega } \) compact, \( \partial \Omega = \gamma \) a piecewise smooth curve, and \( \omega \) a holomorphic differential defined on an open set containing \( \bar{\Omega } ... | Proof. Suitably subdivide \( \Omega \) into a disjoint union \( \Omega = \mathop{\bigcup }\limits_{i}{\Omega }_{i} \) such that for each \( i \), \n\n\[ \n{\Omega }_{i} \subset {U}_{i} \n\]\n\nand \( \partial {\Omega }_{i} \) is a piecewise smooth curve. By using local coordinate representations and applying Cauchy's t... | Yes |
Theorem 5.5 (Stokes’ Theorem for differential forms). Suppose \( C \) is a Riemann surface, \( \Omega \) is an open set in \( C \) with \( \bar{\Omega } \) compact, \( \partial \Omega \) piecewise smooth, and \( \lambda \) is a differential one-form defined on an open set containing \( \bar{\Omega } \) . Then\n\n\[ \n{... | The proof of this theorem can be found in any text containing a discussion of differentiable manifolds. | No |
Theorem 9.6 (Holomorphic implicit function theorem). Suppose \( {f}_{1},\ldots ,{f}_{k} \in {\mathcal{O}}_{n} \) satisfy\n\n\[ \n\det {\left( {\left. \frac{\partial {f}_{i}\left( z\right) }{\partial {z}^{j}}\right| }_{z = 0}\right) }_{1 \leq i, j \leq k} \neq 0.\n\]\n\nThen there exist \( {w}_{1},\ldots ,{w}_{k} \in {\... | Proof. By using the \( {C}^{\infty } \) implicit function theorem, we can get the \( {C}^{\infty } \) functions \( {w}_{1},\ldots ,{w}_{k} \) to satisfy the last condition. All that remains to be proven then is that these \( {w}_{1},\ldots ,{w}_{k} \) are holomorphic.\n\nLet \( {z}^{\prime } = \left( {{z}^{k + 1},\ldot... | Yes |
Proposition 9.7. Suppose \( X \) is a compact, complex manifold and \( Y \) is a closed, connected subset of \( X \) . Then \( Y \) is an \( \left( {n - l}\right) \) -dimensional compact, complex manifold if there exists a covering \( \left\{ {W}_{i}\right\} \) of \( Y \), each \( {W}_{i} \) being a local coordinate ne... | Proof. We only need to exhibit a set of holomorphic local coordinate neighborhoods of \( Y \) . By refining the covering \( \left\{ {W}_{i}\right\} \) if necessary, we may assume that\n\n\[ {W}_{i} = {U}_{i} \times {V}_{i} \subset {\mathbb{C}}^{n - l} \times {\mathbb{C}}^{l} \]\n\nand there exist holomorphic mappings\n... | No |
Proposition 10.5. Suppose \( C \) is a Riemann surface, \( X \) is a complex manifold, \( \varphi \in K\left( X\right) \) is given by \( \left\{ \left( {{U}_{i},{g}_{i},{h}_{i}}\right) \right\} \) and \( f : C \rightarrow X \) is a holomorphic mapping with\n\n\[ f\left( C\right) ⊄ \mathop{\bigcup }\limits_{i}\left\{ {{... | Proof. Setting \( {W}_{i} = {f}^{-1}\left( {U}_{i}\right) \), then \( {f}^{ * }\varphi = \varphi \circ f \) is the meromorphic function given by \( \left\{ \left( {{W}_{i},{f}^{ * }{g}_{i},{f}^{ * }{h}_{i}}\right) \right\} \) so that on \( {W}_{i} \), \n\n\[ {f}^{ * }\varphi = \varphi \circ f = \frac{{g}_{i} \circ f}{{... | Yes |
Corollary 2.4. Theorem 2.2, there exist polynomials \( \alpha ,\beta \in \mathbf{D}\lbrack \lambda \) \( < m \), such that \[ \cdot \mathcal{R}\left( {f, g}\right) \text{.} \] | Proof. In the elim \( \;\therefore \mathcal{H}\left( {f, g}\right) \), denote the cofactor of the element in the \( \left( {m + n}\right) \) th column and \( i \) th row by \( {A}_{i} \), and write \[ \alpha \left( x\right) = {A}_{1}{x}^{n - 1} + \cdots + {A}_{n}, \] \[ \beta \left( x\right) = {A}_{n + 1}{x}^{m - 1} + ... | Yes |
Corollary 2.6. Suppose \( \\mathbf{D} \) is a U.F.D.. Then a necessary and sufficient condition for \( f \\in \\mathbf{D}\\left\\lbrack x\\right\\rbrack \) to have multiple factors is that its discriminant be equal to 0 | \[ \\mathcal{D}\\left( f\\right) = \\mathcal{R}\\left( {f,{f}^{\\prime }}\\right) = 0. \] | Yes |
Theorem 4.5 (Weierstrass preparation theorem). If \( f \in \mathcal{O} \), and \( f\left( {0, y}\right) \) is not identically 0, then inside a suitable neighborhood of \( \left( {0,0}\right) \) , \( f \) has a unique representation\n\n\[ f\left( {x, y}\right) = u\left( {x, y}\right) w\left( {x, y}\right) ,\]\n\nwhere \... | Proof. We have already proven existence in the above, so we shall only prove uniqueness. Since\n\n\[ u\left( {x, y}\right) \neq 0 \]\n\nin a neighborhood of \( \left( {0,0}\right) \), for a fixed value of \( x \) in expression (4.1), \( w\left( {x, y}\right) \) and \( f\left( {x, y}\right) \) have the same zeroes. Thus... | Yes |
Proposition 1.3. If \( D \sim E \), then\n\n\[ \mathcal{L}\left( D\right) \cong \mathcal{L}\left( E\right) ,\;{K}^{1}\left( D\right) \cong {K}^{1}\left( E\right) ,\;\deg D = \deg E. \] | Proof. Since \( D \sim E \), there exists an \( f \in K\left( C\right) \) such that\n\n\[ \left( f\right) = D - E\text{.} \]\n\nNow for any \( g \in \mathcal{L}\left( D\right) \), we have \( \left( g\right) + D \geq 0 \), and thus\n\n\[ \left( {fg}\right) + E = \left( f\right) + \left( g\right) + E = D - E + \left( g\r... | Yes |
Proposition 1.4 (Brill-Noether reciprocity). Suppose \( \omega \in {K}^{1}\left( C\right) \) and \( \left( \omega \right) = D + E \) . Then\n\n\[ \mathcal{L}\left( D\right) \cong {K}^{1}\left( E\right) ,\;\mathcal{L}\left( E\right) \cong {K}^{1}\left( D\right) . \] | Proof. For any \( f \in \mathcal{L}\left( D\right) \), we have \( \left( f\right) \geq - D \), so\n\n\[ \left( {f\omega }\right) = \left( f\right) + \left( \omega \right) \geq - D + D + E = E. \]\n\nWe thus have the mapping\n\n\[ \mathcal{L}\left( D\right) \rightarrow {K}^{1}\left( E\right) \]\n\n\[ f \mapsto {f\omega ... | Yes |
Proposition 2.4. If \( \lambda \) is a closed differential one-form on \( C \), then the mapping\n\n\[ \n{\eta }_{\lambda } : {H}_{1}\left( {C,\mathbb{Z}}\right) \rightarrow \mathbb{C},\n\]\n\n\[ \n\left\lbrack \gamma \right\rbrack \mapsto {\int }_{\gamma }\lambda \n\]\n\nwhere \( \gamma \) is a closed curve on \( C \)... | Proof. If \( {\gamma }^{\prime } \) is homologous to \( \gamma \), then \( \gamma - {\gamma }^{\prime } = \partial \Omega \), where \( \Omega \) is a region on \( C \) . By Stokes’ theorem, we have\n\n\[ \n{\eta }_{\lambda }\left( \gamma \right) - {\eta }_{\lambda }\left( {\gamma }^{\prime }\right) = {\int }_{\gamma - ... | Yes |
Proposition 2.5. If \( \lambda \) is a closed differential one-form on \( C \), and if for all \( i \) we have\n\n\[ \n{\int }_{{\gamma }_{i}}\lambda = 0 \n\]\n\nthen \( \lambda \) is exact. | Proof. Fix the point \( {p}_{0} \) on \( C \) . For any point \( p \) on \( C \), choose a path \( \gamma \) from \( {p}_{0} \) to \( p \), and define\n\n\[ \nf\left( p\right) = {\int }_{\gamma }\lambda \n\]\n\nWe must ascertain that the definition of \( f\left( p\right) \) is independent of the choice of the path \( \... | Yes |
Proposition 2.6. Suppose \( C \) is a compact Riemann surface, and \( \omega ,\varphi \in \) \( {\Omega }^{1}\left( C\right) \) . Considering \( \omega \) and \( \varphi \) as differential one-forms on \( C \), if\n\n\[ \omega + \bar{\varphi } = {df} \]\n\nwhere \( f \) is a \( {C}^{\infty } \) function on \( C \), the... | Proof. Suppose \( \omega = h\left( z\right) {dz},\varphi = g\left( z\right) {dz} \) . Then\n\n\[ \omega \land \varphi = 0, \]\n\nand\n\n\[ \frac{i}{2}\varphi \land \bar{\varphi } = {\left| g\left( z\right) \right| }^{2}\frac{i}{2}{dz} \land d\bar{z} = {\left| g\left( z\right) \right| }^{2}{du} \land {dv}. \]\n\nWe shal... | Yes |
Proposition 2.7. Under the hypothesis of Theorem 2.1, we have\n\n\[ \dim {\Omega }^{1}\left( C\right) \leq g \] | Proof. Assuming \( \dim {\Omega }^{1}\left( C\right) \geq g + 1 \), let us suppose that the \( \left( {g + 1}\right) \) elements \( {\omega }_{1},\ldots ,{\omega }_{g + 1} \in {\Omega }^{1}\left( C\right) \) are linearly independent over \( \mathbb{C} \), and consider the equations\n\n\[ {\int }_{{\gamma }_{i}}\mathop{... | Yes |
Proposition 3.6. The \( {2g} \) period vectors given above are \( \mathbb{R} \) -linearly independent. | Proof. We prove this by contradiction. Suppose \( {\pi }_{1},\ldots ,{\pi }_{2g} \) are \( \mathbb{R} \) - dependent. Then there exist real numbers, \( {a}_{1},\ldots ,{a}_{2g} \), not all zero, such that\n\n\[ \n{a}_{1}{\pi }_{1} + \cdots + {a}_{2g}{\pi }_{2g} = 0.\n\]\n\n(3.1)\n\nTaking the complex conjugate of this,... | Yes |
Corollary 4.2. If \( d \geq g \), then\n\n\[ l\left( D\right) \neq 0.\text{.} \] | In other words, if \( l\left( D\right) = 0 \), then\n\n\[ d \leq g - 1.\text{.} \] | No |
Proposition 4.3. With the hypothesis of Proposition 4.1, we have\n\n\[ i\left( D\right) \geq g - d - 1 \] | Proof. We shall use the notation in the proof of Proposition 4.1, viz., \( F, m,{D}^{\prime },{D}^{\prime \prime },{d}^{\prime },{d}^{\prime \prime } \), and \( \Delta \) .\n\nFirst, we select a homogeneous polynomial \( G\left( {{\xi }^{0},{\xi }^{1},{\xi }^{2}}\right) \) of degree \( n \) such that \( F \nmid G \) an... | Yes |
Proposition 1.1. Suppose \( C \) is a compact Riemann surface of genus 0. Then \( C \cong {\mathbb{P}}^{1} \) . | Proof. Choose a point \( p \) on \( C \) and let \( D = p \in \operatorname{Div}\left( C\right) \) . We have\n\n\[ 0 \leq i\left( D\right) \leq \dim {\Omega }^{1}\left( C\right) = g = 0.\]\n\nThus \( i\left( D\right) = 0 \), and by the Riemann-Roch theorem, we get\n\n\[ l\left( D\right) = d - g + i\left( D\right) + 1 =... | Yes |
Proposition 2.1. Suppose \( C \) is a compact Riemann surface of genus 1. Then \( C \) can be represented by a smooth algebraic curve of degree 3 in \( {\mathbb{P}}^{2} \) . | Proof. Our aim is to construct a holomorphic injective mapping \( f \) from \( C \) into \( {\mathbb{P}}^{2} \) such that \( f\left( C\right) \) is a smooth algebraic curve of degree 3 .\n\nWe first point out that for any nonzero \( \omega \in {\Omega }^{1}\left( C\right) \) and any point \( p \in C \), it is always th... | Yes |
Proposition 3.4. \( {\varphi }_{K} \) is nondegenerate. (See Theorem 10.1 in Chapter I.) | Proof. Let us assume that \( {\varphi }_{K} \) is degenerate. Then there exist \( {\lambda }_{\alpha } \in \mathbb{C} \) \( \left( {\alpha = 1,2,\ldots, g}\right) \), not all zero, such that for all \( p \in C \) we have\n\n\[ \mathop{\sum }\limits_{{\alpha = 1}}^{g}{\lambda }_{\alpha }{\omega }_{\alpha }\left( p\right... | Yes |
Proposition 5.1. All compact Riemann surfaces of genus 2 are hyperelliptic. | Proof. Suppose \( C \) is a compact Riemann surface of genus 2. According to Proposition 3.5, it suffices to prove that the canonical map \( {\varphi }_{K} \) on \( C \) is not injective. In fact, if \( {\varphi }_{K} : C \rightarrow {\mathbb{P}}^{1} \) is injective, then \( C \) has genus zero. Q.E.D. | No |
Proposition 6.5. Suppose \( C \) is a canonical curve of genus \( g \) . Then\n\n\[ \deg C = {2g} - 2\text{.} \] | Proof. Suppose \( {\omega }_{1},{\omega }_{2},\ldots ,{\omega }_{g} \) form a basis of \( {\Omega }^{1}\left( C\right) \) . Then\n\n\[ C = \left\{ {\left\lbrack {{\omega }_{1}\left( p\right) ,\ldots ,{\omega }_{g}\left( p\right) }\right\rbrack, p \in C}\right\} ,\]\n\nand suppose\n\n\[ H = \left\{ {\mathop{\sum }\limit... | Yes |
Proposition 6.6. A canonical curve of genus 3 is a smooth plane algebraic curve of degree 4. | Proof. By definition, a canonical curve of genus 3 must lie in \( {\mathbb{P}}^{2} \) . We have already proven its smoothness (see Proposition 3.8), and by the preceding proposition, we have\n\n\[ \deg C = {2g} - 2 = 2 \times 3 - 2 = 4\text{. Q.E.D.} \] | Yes |
Proposition 1.6. \( \operatorname{Im}\left( \right) \subset \ker u \), i.e., for any \( f \in {K}^{ * }\left( C\right) \) with \( \left( f\right) = D \) , then | \[ u\left( D\right) = 0\text{.} \] | No |
Proposition 1.7. \( \ker u \subset \operatorname{Im}\left( \right) \), i.e., if \( u\left( D\right) = 0 \) where \( D \in {\operatorname{Div}}^{0}\left( C\right) \) , there exists \( f \in {K}^{ * }\left( C\right) \) such that | \[ \left( f\right) = D\text{.} \] | No |
Proposition 2.1. Given \( \varphi \in {K}^{1}{\left( C\right) }^{ - } \) which satisfies (2.1). Suppose \( q \) is a fixed point on \( C \), and let\n\n\[ f\left( p\right) = \exp \left( {2\sqrt{-1}\pi {\int }_{q}^{p}\varphi }\right) \]\n\nwhere the integration is along any path that does not pass through a pole of \( \... | Proof. Given any two paths from \( q \) to \( p \) which do not pass through a pole of \( \varphi \), let\n\n\[ {\int }_{q}^{p}\varphi \;\text{ and }\;{\int }_{q}^{p}\varphi \]\n\nrepresent their respective integrals. Since \( \varphi \) satisfies (2.1), \( {\int }_{q}^{p}\varphi - {}^{\prime }{\int }_{q}^{p}\varphi \)... | Yes |
Proposition 4.2. \( {C}^{\left( d\right) } \) is in a natural way a complex manifold. | Proof. Suppose \( {C}^{d} = C \times \cdots \times C \) is the \( d \) -fold direct product of \( C \) with itself. It is a complex manifold.\n\nDenote by \( {\sum }_{d} \) the permutation group on \( \{ 1,\cdots, d\} \) . Then for any \( \sigma \in {\sum }_{d} \), we define \( \sigma : {C}^{d} \rightarrow {C}^{d} \) b... | No |
Proposition 4.7. If \( D \) is a generic divisor, then\n\n\[ \operatorname{rank}{\left( {u}_{ * }\right) }_{D} = \dim \overline{{\varphi }_{K}\left( D\right) } + 1 \] | Proof. From the Brill-Noether matrix we see that \( \operatorname{rank}{\left( {u}_{ * }\right) }_{D} \) is just the dimension of the linear space spanned by \( {\varphi }_{K}\left( {p}_{i}\right) \;\left( {i = 1,\cdots, d}\right) \) , and by definition \( \dim \overline{{\varphi }_{K}\left( D\right) } \) is also the d... | Yes |
Corollary 4.8. Suppose \( g \geq 1 \) . Then generically on \( {C}^{\left( g\right) } \) we have\n\n\[ \operatorname{rank}{u}_{ * } = g. \] | Proof. If \( g = 1 \), obviously,\n\n\[ {\left( {u}_{ * }\right) }_{p} = \omega \left( p\right) \neq 0. \]\n\nWe therefore always have\n\n\[ \operatorname{rank}\left( {{u}_{ * }\left( p\right) }\right) = 1. \]\n\nSuppose now that \( g \geq 2 \), and let \( D = {p}_{1} + \cdots + {p}_{g} \in {C}^{\left( g\right) } \) be... | Yes |
Proposition 4.9. The restriction of the Abel-Jacobi map to \( {C}^{\left( g\right) } \)\n\n\[ u : {C}^{\left( g\right) } \rightarrow J\left( C\right) ,\]\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{g}{p}_{i} \mapsto \left( {\mathop{\sum }\limits_{{i = 1}}^{g}{\int }_{q}^{{p}_{i}}{\omega }_{1},\cdots ,\mathop{\sum }\limits_{... | Proof of Proposition 4.9. It suffices to show that the map\n\n\[ u : {C}^{\left( g\right) } \rightarrow J\left( C\right) \]\n\nfulfills the two conditions of Fact 4.10. Certainly \( {C}^{\left( g\right) } \) and \( J\left( C\right) \) are both complex manifolds,\n\n\[ \dim {C}^{\left( g\right) } = g = \dim J\left( C\ri... | Yes |
Theorem 1.1.1. Let \( p \) be an odd prime number, then there exist \( a, b \in \mathbb{Z} \) such that \( p = \) \( {a}^{2} + {b}^{2} \) if and only if \( p \equiv 1\\left( {\\operatorname{mod}\;4}\\right) \) . | The direction \ | No |
Proposition 1.1.4. If \( R \) is euclidean, then \( R \) is a principal ideal domain. | Proof. Using the conditions \( 1\& 2 \), one sees \( R \) is a domain. Let \( I \) be an ideal of \( R \), and let \( \beta \in I \) such that \( f\left( \beta \right) = \mathop{\min }\limits_{{0 \neq \alpha \in I}}f\left( x\right) \) . For \( \alpha \in I \), there exists \( \delta \) such that \( \alpha = {\beta \gam... | Yes |
Proposition 1.1.5. \( \mathbb{Z}\left\lbrack i\right\rbrack \) is euclidean. | Proof. Put \( f : \mathbb{Z}\left\lbrack i\right\rbrack \rightarrow {\mathbb{Z}}_{ \geq 0}, a + {bi} \mapsto {a}^{2} + {b}^{2} \) . We check \( f \) satisfies the conditions in the definition. The conditions 1 and 2 are clear. Let \( \alpha ,\beta \in \mathbb{Z}\left\lbrack i\right\rbrack ,\beta \neq 0 \) . Consider\n\... | Yes |
Proposition 1.1.7. There exist \( a, b \in \mathbb{Z} \) such that \( p = {a}^{2} + {b}^{2} \) if and only if \( \left( p\right) \) is not a prime ideal in \( \mathbb{Z}\left\lbrack i\right\rbrack \) . (Recall an ideal \( I \) is called a prime ideal if \( I \supset {I}_{1}{I}_{2} \Rightarrow I \supset {I}_{1} \) or \(... | Proof. If \( p = {a}^{2} + {b}^{2} \), then \( p = \left( {a + {bi}}\right) \left( {a - {bi}}\right) \) . If \( \left( p\right) \) is a prime ideal then \( \left( p\right) \supset \left( {a + {bi}}\right) \) or \( \left( p\right) \supset \left( {a - {bi}}\right) \) . Replacing \( b \) by \( - b \) if needed, we assume ... | Yes |
Proposition 1.2.4. Let \( A \subset B \) be commutative rings with 1, let \( b \in B \) . Then \( b \) is integral over \( A \) if and only if there exists an \( A \) -subalgebra \( C \) of \( B \) that is finitely generated as \( A \) -module such that \( A\left\lbrack b\right\rbrack \subseteq C \) . | Proof. \ | No |
Corollary 1.2.5. Let \( A \hookrightarrow B \) be commutative rings with 1, let \( \alpha ,\beta \in B \) be integral over A. Then \( \alpha + \beta ,{\alpha \beta } \) are also integral over \( A \) . | Proof. Since \( \beta \) is integral over \( A,\beta \) is integral over \( A\left\lbrack \alpha \right\rbrack \) . By the above proposition (and the proof), \( A\left\lbrack \alpha \right\rbrack \left\lbrack \beta \right\rbrack \) is a fintiely generated \( A\left\lbrack \alpha \right\rbrack \) -module. Since \( \alph... | Yes |
Corollary 1.2.7. Let \( A \subset B \subset C \) be commutative rings with 1 . If \( C \) is integral over \( B \) , and \( B \) is integral over \( A \), then \( C \) is integral over \( A \) . | Proof. Let \( c \in C \), there exist thus \( {b}_{0},\cdots ,{b}_{n - 1} \in B \) such that\n\n\[ \n{c}^{n} + {b}_{n - 1}{c}^{n - 1} + \cdots + {b}_{0} = 0.\n\]\n\nThis implies that \( A\left\lbrack {{b}_{0},\cdots ,{b}_{n - 1}}\right\rbrack \left\lbrack C\right\rbrack \) is a finitely generated \( A\left\lbrack {{b}_... | Yes |
Lemma 1.2.10. A UFD \( R \) is integrally closed. | Proof. Let \( K \mathrel{\text{:=}} \operatorname{Frac}\left( R\right) \), and let \( \alpha = \frac{a}{b} \in K \) with \( \left( {a, b}\right) = 1 \) . Suppose \( \alpha \) is integral over \( R \), then there exists \( {c}_{0},\cdots ,{c}_{n - 1} \in R \) such that\n\n\[{\left( \frac{a}{b}\right) }^{n} + {c}_{n - 1}... | Yes |
Proposition 1.2.14. Let \( A \) be an integral closed domain, \( K \mathrel{\text{:=}} \operatorname{Frac}\left( A\right) \) . Let \( L \) be a finite extension of \( K, B \) be the integral closure of \( A \) in \( L \) . Then \( b \in B \) if and only if the monic minimal polynomial of \( b \) over \( K \) has coeffi... | Proof. The \ | No |
Proposition 1.3.1. Let \( L/K \) be a separable finite extension, \( {\sum }_{L} \mathrel{\text{:=}} \left\{ {\sigma : L \hookrightarrow \bar{K}{\left| \sigma \right| }_{K} = }\right. \) id \( \} \) . Then we have\n\n1. \( {f}_{x}\left( T\right) = \mathop{\prod }\limits_{{\sigma \in {\sum }_{L}}}\left( {T - \sigma \lef... | Proof. Consider \( K \subset K\left( x\right) \subset L \) . Let \( p\left( t\right) \) be the minimal polynomial of \( x \) over \( K,{d}_{1} \mathrel{\text{:=}} \) \( \deg \left( {p\left( t\right) }\right) \) . Thus \( K\left( x\right) = K \oplus {Kx} \oplus \cdots \oplus K{x}^{{d}_{1} - 1} \) . Let \( {\left\{ {e}_{... | Yes |
Corollary 1.3.2. Let \( K \subset L \subset M \) be finite separable extensions, then \( {\operatorname{Tr}}_{M/K} = {\operatorname{Tr}}_{L/K} \circ {\operatorname{Tr}}_{M/L} \) , \( {N}_{M/K} = {N}_{L/K} \circ {N}_{M/L} \) | Proof. Exercise. | No |
Corollary 1.3.4. Let \( A \subset K \) be integral closed, \( B \subset L \) be the integral closure of \( A \) in \( L \) , then for any \( x \in B,{\operatorname{Tr}}_{L/K}\left( x\right) \in A \), and \( {N}_{L/K}\left( x\right) \in A \) . | Proof. Let \( p\left( T\right) \in A\left\lbrack T\right\rbrack \) be the minimal polynomial of \( x \) over \( K, r \mathrel{\text{:=}} \deg p\left( T\right) = \left\lbrack {K\left( x\right) : K}\right\rbrack \) . Thus \( \left\{ {1, x,\cdots ,{x}^{r - 1}}\right\} \) is a basis of \( K\left( x\right) \) over \( K \) .... | Yes |
Proposition 1.3.5. Suppose \( L/K \) is separable, then the above pairing \( \langle \) , \( \rangle {isnon} \) -degenerate, i.e. for \( x \in L \), if \( {\operatorname{Tr}}_{L/K}\left( {xy}\right) = 0 \) for all \( y \in L \), then \( x = 0 \) . | Proof. Fact (linear algebra): \( \langle \) , \( \rangle {isnon} \) -degenerate if and only for any basis \( {e}_{1},\cdots ,{e}_{d} \) of \( L \) over \( K \), the matrix \( {\left( {\operatorname{Tr}}_{L/K}\left( {e}_{i}{e}_{j}\right) \right) }_{1 \leq i, j \leq d} \in {M}_{d}\left( K\right) \) is invertible.\n\nLet ... | Yes |
Proposition 1.3.7. Let \( K/\mathbb{Q} \) be a number field, \( d = \left\lbrack {K : \mathbb{Q}}\right\rbrack \) . Then \( {\mathcal{O}}_{K} \) is a free \( \mathbb{Z} \) -module of rank \( d \) . | Proof. By the exercise, there exist \( {e}_{1},\cdots ,{e}_{d} \in {\mathcal{O}}_{K} \) such that \( \left\{ {e}_{i}\right\} \) form a basis of \( K \) over \( \mathbb{Q} \), and that\n\n\[ M \mathrel{\text{:=}} \mathbb{Z}{e}_{1} \oplus \cdots \oplus \mathbb{Z}{e}_{d} \subseteq {\mathcal{O}}_{K} \]\n\nLet \( \widetilde... | Yes |
Consider \( K = \mathbb{Q}\left( \sqrt{-5}\right) \), in this case \( {\mathcal{O}}_{K} = \mathbb{Z} \oplus \mathbb{Z}\sqrt{-5} \) . In \( {\mathcal{O}}_{K} \), we have \[ {21} = 3 \cdot 7 = \left( {1 + 2\sqrt{-5}}\right) \left( {1 - 2\sqrt{-5}}\right) . \] We claim \( 3,7,\left( {1 + 2\sqrt{-5}}\right) ,\left( {1 - 2\... | if not, write \( 1 + 2\sqrt{-5} = {\alpha \beta } \), with \( \alpha ,\beta \) non-unit. We have \( {N}_{K/\mathbb{Q}}\left( {1 + 2\sqrt{-5}}\right) = \left( {1 + 2\sqrt{-5}}\right) \left( {1 - 2\sqrt{-5}}\right) = {21} \) . If \( {N}_{K/\mathbb{Q}}\left( \alpha \right) = \alpha {\alpha }^{c} = 1 \), then \( \alpha \) ... | Yes |
Proposition 1.4.3. Let \( A \) be a noetherian (commutative) ring, \( M \) is a finitely generated A-module. Then any submodule of \( M \) is finitely generated. | Proof. We run an induction on the numbers of generators of \( M \) .\n\nIf \( M \) can be generated by one element \( e \) . Then we have a surjective map \( f : A \rightarrow M \) , \( a \mapsto {ae} \) . For any submodule \( N \) of \( M,{f}^{-1}\left( N\right) \) is an ideal of \( A \), hence is finitely generated. ... | Yes |
Theorem 1.4.4. Let \( K \) be a number field. Then \( {\mathcal{O}}_{K} \) is a Dedekind domain, i.e. integrally closed, noetherian, and any non-zero prime ideal of \( {\mathcal{O}}_{K} \) is maximal. | Proof. We have seen that \( {\mathcal{O}}_{K} \) is integrally closed.\n\nLet \( I \) be a non-zero ideal of \( {\mathcal{O}}_{K} \), then there exists \( 0 \neq n \in \mathbb{Z} \) such that \( n \in I \) (e.g. let \( \alpha \in I \), then we can take \( \left. {n \mathrel{\text{:=}} {N}_{K/\mathbb{Q}}\left( \alpha \r... | Yes |
Lemma 1.4.7. For \( 0 \neq \mathfrak{a} \subset {\mathcal{O}}_{K} \), the following holds:\n\n\[ \exists {\mathfrak{p}}_{1},\cdots ,{\mathfrak{p}}_{r}\text{prime ideals of}{\mathcal{O}}_{K}\text{such that}\mathfrak{a} \supset {\mathfrak{p}}_{1}\cdots {\mathfrak{p}}_{r}\text{.} \] | Proof. Let \( S \) be the set: \( \left\{ {I\text{non-zero proper ideal of}{\mathcal{O}}_{K}, I}\right. \) does not satisfy (1.1) \( \} \) . Suppose the set is non-empty. Since \( {\mathcal{O}}_{K} \) is noetherian, there exists a maximal element \( \mathfrak{a} \) in the set. It is clear that \( \mathfrak{a} \) is not... | Yes |
Lemma 1.4.8. Let \( \mathfrak{a} \) be a non-zero ideal, then \( \mathfrak{a}{\mathfrak{p}}^{-1} \mathrel{\text{:=}} \left\{ {\mathop{\sum }\limits_{i}{a}_{i}{x}_{i} \mid {a}_{i} \in \mathfrak{a},{x}_{i} \in {\mathfrak{p}}^{-1}}\right\} \neq \mathfrak{a} \) . | Proof. We first show \( {\mathfrak{p}}^{-1} \neq {\mathcal{O}}_{K} \) . Let \( 0 \neq a \in \mathfrak{p} \), by the previous lemma, there exist \( {\mathfrak{p}}_{1},\cdots ,{\mathfrak{p}}_{r} \) such that\n\n\[ \mathfrak{p} \supset \left( a\right) \supset {\mathfrak{p}}_{1}\cdots {\mathfrak{p}}_{r} \]\n\nWe choose \( ... | No |
Lemma 1.4.12. Let \( \mathfrak{a} \) be a fractional ideal, then there exists \( c \in {K}^{ \times },{\mathfrak{a}}_{0} \subset {\mathcal{O}}_{K} \) such that \( \mathfrak{a} = c{\mathfrak{a}}_{0} \) | Proof. Suppose \( \mathfrak{a} \) is generated by \( {e}_{1},\cdots ,{e}_{n} \) . Let \( c \in {K}^{ \times } \) such that \( \frac{1}{c}{e}_{i} \in {\mathcal{O}}_{K} \) . Thus \( {\mathfrak{a}}_{0} \mathrel{\text{:=}} \frac{1}{c}\mathfrak{a} \) is an ideal of \( {\mathcal{O}}_{K} \) and \( \mathfrak{a} = c\left( {\fra... | Yes |
Lemma 1.4.13. Let \( \mathfrak{a} \) be a fractional ideal, then \( \mathfrak{a}{\mathfrak{a}}^{-1} = {\mathcal{O}}_{K} \) . | Proof. For \( c \in {K}^{ \times } \), it is easy to check \( {\left( c\mathfrak{a}\right) }^{-1} = {c}^{-1}{\mathfrak{a}}^{-1} \) . Together with the previous lemma, it suffices to prove the lemma for ideal \( \mathfrak{a} \subset {\mathcal{O}}_{K} \) . Using the unique prime factorization, we write \( \mathfrak{a} \)... | Yes |
Proposition 1.4.14. Every fractional ideal \( \mathfrak{a} \) admits a unique factorization \( \mathfrak{a} = \mathop{\prod }\limits_{\mathfrak{p}}{\mathfrak{p}}^{{v}_{\mathfrak{p}}} \) , where \( {v}_{\mathfrak{p}} \in \mathbb{Z} \), and \( {v}_{\mathfrak{p}} = 0 \) for all but finitely many prime ideals \( \mathfrak{... | Proof. Suppose \( \mathfrak{a} = \frac{1}{c}{\mathfrak{a}}_{0} \), with \( {\mathfrak{a}}_{0} \subseteq {\mathcal{O}}_{K} \) and \( c \in {\mathcal{O}}_{K} \) . We have \( \left( c\right) = \mathop{\prod }\limits_{\mathfrak{p}}{\mathfrak{p}}^{{v}_{\mathfrak{p}}\left( c\right) } \) and \( {\mathfrak{a}}_{0} = \mathop{\p... | Yes |
Lemma 1.5.1. Let \( 0 \neq \mathfrak{a} \subseteq {\mathcal{O}}_{K},{e}_{1},\cdots ,{e}_{d} \) be a basis of \( {\mathcal{O}}_{K} \) over \( \mathbb{Z},{f}_{1},\cdots ,{f}_{d} \) be a basis of \( \mathfrak{a} \) over \( \mathbb{Z} \), and let \( A \in {M}_{n}\left( \mathbb{Z}\right) \) such that \( \left( {{f}_{1},\cdo... | Proof. By the structure theorem of abelian groups, there exist a basis \( {e}_{1}^{\prime },\cdots ,{e}_{d}^{\prime } \) of \( {\mathcal{O}}_{K} \) over \( \mathbb{Z} \) and \( {a}_{1},\cdots ,{a}_{d} \in {\mathbb{Z}}_{ \geq 1} \) such that \( \left\{ {{a}_{i}{e}_{i}}\right\} \) is a basis of \( \mathfrak{a} \) over \(... | Yes |
Lemma 1.5.2. Suppose \( \mathfrak{a} = {\mathfrak{p}}_{1}^{{e}_{1}}\cdots {\mathfrak{p}}_{r}^{{e}_{r}} \), then \( N\left( \mathfrak{a}\right) = \mathop{\prod }\limits_{{i = 1}}^{r}N{\left( {\mathfrak{p}}_{i}\right) }^{{e}_{i}} \) . | Proof. By Chinese reminder theorem, we have \( {\mathcal{O}}_{K}/\mathfrak{a} \cong \mathop{\prod }\limits_{{i = 1}}^{r}{\mathcal{O}}_{K}/{\mathfrak{p}}_{i}^{{e}_{i}} \). We reduce to show that for a prime ideal \( \mathfrak{p}, N\left( {\mathfrak{p}}^{e}\right) = N{\left( \mathfrak{p}\right) }^{e} \). Consider the exa... | Yes |
Lemma 1.5.4. For a prime number \( p \), there are finitely many prime ideals \( \mathfrak{p} \subset {\mathcal{O}}_{K} \) such that \( p \in \mathfrak{p} \) . | Proof. Applying prime factorization to the ideal \( p{\mathcal{O}}_{K} : p{\mathcal{O}}_{K} = {\mathfrak{p}}_{1}\cdots {\mathfrak{p}}_{r} \) . Then \( p \in \mathfrak{p} \Leftrightarrow \) \( \mathfrak{p} = {\mathfrak{p}}_{i} \) for some \( i \) (Or use the fact that \( {\mathcal{O}}_{K}/p \) has finite cardinality hen... | Yes |
Corollary 1.5.5. Let \( M \in {\mathbb{Z}}_{ \geq 1} \), there exist only finitely many ideals \( \mathfrak{a} \) such that \( N\left( \mathfrak{a}\right) \leq M \) . | Proof. By Lemma 1.5.2, Lemma 1.5.4 and the discussion above it, the corollary follows from the fact that the set \( \left\{ {{p}_{1}^{{n}_{1}}\cdots {p}_{r}^{{n}_{r}} \mid {p}_{i}}\right. \) are prime numbers and \( \left. {{n}_{i} > 0}\right\} \) is finite. | No |
Theorem 1.5.6. The class group \( {C}_{K} \) is finite. | To prove the theorem, we will show the following statement:\n\n- there exists \( M > 0 \) such that for any ideal \( \mathfrak{a} \subset {\mathcal{O}}_{K} \), there exists \( \alpha \in \mathfrak{a} \) with \( N\left( \alpha \right) \leq \) \( {MN}\left( \mathfrak{a}\right) \) .\n\nActually, if this holds, then we hav... | Yes |
Proposition 1.6.2. A subgroup \( \Lambda \) of \( V \) (equipped with the standard topology of \( {\mathbb{R}}^{n} \) ) is a lattice if and only if \( \Lambda \) is discrete, i.e. for all \( \gamma \in \Lambda \), there exists an open neighborhood \( U \) of \( \gamma \) such that \( U \cap \Lambda = \{ \gamma \} \) . | Proof. \ | No |
Lemma 1.6.3. A lattice \( \Lambda \subset V \) is complete if and only if there exists a bounded subset \( M \subset V \) such that \( V = { \cup }_{\gamma \in \Gamma }\left( {\gamma + M}\right) \) . | Proof. \ | No |
Lemma 1.6.4. Let \( {v}_{1},\cdots ,{v}_{n} \) be another basis of \( V \), and let \( A \in {\mathrm{{GL}}}_{n}\left( \mathbb{R}\right) \) such that\n\n\[ \left( {{v}_{1},\cdots ,{v}_{n}}\right) = \left( {{e}_{1},\cdots ,{e}_{n}}\right) A.\]\n\nLet \( \Phi \) be the fundamental mesh of \( \Lambda = \mathbb{Z}{v}_{1} +... | Proof. Writting \( A \) as a product of elementary matrices, we reduce to prove the lemma in the case where \( A \) is an elementary matrice. However, this case is clear. | No |
Theorem 1.6.5 (Minkowski’s lattice point theorem). Let \( \Lambda \) be a complete lattice in \( V, X \) is a centrally symmetric (i.e. \( x \in X \Leftrightarrow - x \in X \) ), convex subset of \( V \) (i.e. if \( x, y \in X \) , then \( {tx} + \left( {1 - t}\right) y \in X \) for all \( 0 \leq t \leq 1 \) ). If \( \... | Proof. It suffices to show there exist \( {\gamma }_{1} \neq {\gamma }_{2} \in \Gamma \) such that \( \left( {{\gamma }_{1} + \frac{1}{2}X}\right) \cap \left( {{\gamma }_{2} + \frac{1}{2}X}\right) \neq \varnothing \) . Indeed, if so, there exist \( {x}_{1},{x}_{2} \in X \) such that \( \frac{1}{2}{x}_{1} + {\gamma }_{1... | Yes |
Theorem 1.7.3. Let \( \mathfrak{a} \) be a non-zero ideal of \( {\mathcal{O}}_{K} \), let \( {c}_{\sigma } > 0 \) for all \( \sigma \in {\sum }_{\infty } \) such that \( {c}_{\bar{\sigma }} = {c}_{\sigma } \) and \[ \mathop{\prod }\limits_{\sigma }{c}_{\sigma } > {\left( \frac{2}{\pi }\right) }^{s}\sqrt{\left| {\Delta ... | Proof. Let \( X \mathrel{\text{:=}} \left\{ {\left( {z}_{\sigma }\right) \in {K}_{\mathbb{R}}\left| \right| {z}_{\sigma } \mid < {c}_{\sigma }}\right\} \) . It is clear that \( X \) is centrally symmetric, and convex in \( {K}_{\mathbb{R}} \) . By calculation, we have \( {\operatorname{Vol}}_{1}\left( X\right) = {2}^{r... | No |
Corollary 1.7.4. For a non-zero ideal \( \mathfrak{a} \), there exists \( \alpha \in \mathfrak{a}, N\left( {\alpha {\mathcal{O}}_{K}}\right) \leq {\left( \frac{2}{\pi }\right) }^{s}\sqrt{\left| {\Delta }_{K}\right| }N\left( \mathfrak{a}\right) \) . | Proof. For any \( \epsilon > 0 \), the above theorem implies that there exists \( 0 \neq \alpha \in \mathfrak{a} \) such that \( N\left( \alpha \right) < {\left( \frac{2}{\pi }\right) }^{s}\sqrt{\left| {\Delta }_{K}\right| }N\left( \mathfrak{a}\right) + \epsilon \) . Together with the fact that \( N\left( \alpha \right... | Yes |
Lemma 1.8.1. We have \( \operatorname{Ker}\left( \lambda \right) = \mu \left( {\mathcal{O}}_{K}\right) = \left\{ \right. \) roots of unity in \( \left. {\mathcal{O}}_{K}\right\} \), that is a finite group. | Proof. By definition, \( x \in \operatorname{Ker}\left( \lambda \right) \Leftrightarrow \left| {\sigma \left( x\right) }\right| = 1 \) for all \( \sigma \in {\sum }_{\infty } \) It is then clear that any root of unity in \( {\mathcal{O}}_{K} \) is contained in \( \operatorname{Ker}\left( \lambda \right) \) . Consider t... | Yes |
Proposition 1.8.2. \( \Lambda \) is a lattice. | Proof. It suffices to show \( \Lambda \) is discrete. Take \( U \mathrel{\text{:=}} \left\{ {\left( {{x}_{1},\cdots ,{x}_{r + s}}\right) \in {\mathbb{R}}^{r + s}\left| \right| {x}_{i} \mid \leq t}\right\} \), then \( {\ell }^{-1}\left( U\right) = \left\{ {\left( {{x}_{1},\cdots ,{x}_{r + {2s}}}\right) \in {K}_{\mathbb{... | Yes |
Lemma 1.8.3. For \( a \in {\mathbb{Z}}_{ > 0} \), there are only finitely many, up to multiplication by elements in \( {\mathcal{O}}_{K}^{ \times },\alpha \in {\mathcal{O}}_{K} \), such that \( N\left( \alpha \right) \mathrel{\text{:=}} \left| {{N}_{K/\mathbb{Q}}\left( \alpha \right) }\right| = a \) . | Proof. Recall \( \left| {{N}_{K/\mathbb{Q}}\left( \alpha \right) }\right| = N\left( {\alpha {\mathcal{O}}_{K}}\right) \) . Recall there are only finitely many integral ideal \( \mathfrak{a} \) such that \( N\left( \mathfrak{a}\right) = a \) . Note also that if \( \alpha {\mathcal{O}}_{K} = \beta {\mathcal{O}}_{K} \), t... | No |
Proposition 1.9.2. We have \( \mathop{\sum }\limits_{{i = 1}}^{g}{e}_{i}{f}_{i} = d = \left\lbrack {L : K}\right\rbrack \) . | Proof. We have \( \left| {{\mathcal{O}}_{L}/\mathfrak{p}{\mathcal{O}}_{L}}\right| = \mathop{\prod }\limits_{{i = 1}}^{g}{\left| {\mathcal{O}}_{L}/{\mathfrak{P}}_{i}\right| }^{{e}_{i}} \), and \( \left| {{\mathcal{O}}_{L}/{\mathfrak{P}}_{i}}\right| = {\left| {\mathcal{O}}_{K}/fp\right| }^{{f}_{i}} \). It suffices to sho... | No |
Lemma 1.9.4. \( \left| {{\mathcal{O}}_{L}/{\mathcal{O}}_{K}\left\lbrack \theta \right\rbrack }\right| \) is finite. | Proof. Let \( {\alpha }_{1},\cdots ,{\alpha }_{m} \) be a set of generators of \( {\mathcal{O}}_{L} \) over \( {\mathcal{O}}_{K} \) . Since \( {\alpha }_{i} \in L = K\left( \theta \right) \), there exists \( {a}_{i} \in {\mathcal{O}}_{K} \smallsetminus \{ 0\} \) such that \( {a}_{i}{\alpha }_{i} \in {\mathcal{O}}_{K}\l... | Yes |
Proposition 1.9.5. Let \( \mathfrak{p} \) be a prime ideal of \( {\mathcal{O}}_{K} \), and suppose \( \mathfrak{p} \) is relatively prime to \( {\mathcal{F}}_{\theta } \) . Let \( \bar{p}\left( x\right) \mathrel{\text{:=}} {\bar{p}}_{1}{\left( x\right) }^{{e}_{1}}\cdots {\bar{p}}_{g}{\left( x\right) }^{{e}_{r}} \) be t... | Proof. Consider the natural morphism of \( {\mathcal{O}}_{K} \) -algebras \( f : {\mathcal{O}}_{K}\left\lbrack \theta \right\rbrack /\mathfrak{p} \rightarrow {\mathcal{O}}_{L}/\mathfrak{p} \) . We show \( f \) is an isomorphism. Since \( \mathfrak{p} \) is relatively prime to \( {\mathcal{F}}_{\theta } \), we have\n\n\... | Yes |
Lemma 1.9.6. Keep the above notation, let \( \mathfrak{p} \) be a non-zero prime ideal of \( {\mathcal{O}}_{K} \). Then \( \mathfrak{p} \mid d\left( {{\mathcal{O}}_{K}\left\lbrack \theta \right\rbrack }\right) \) if and only if the mod \( \mathfrak{p} \) reduction \( \bar{p}\left( x\right) \in k\left\lbrack x\right\rbr... | Proof. Let \( M \) be the Galois closure of \( L \) over \( K \). In \( {\mathcal{O}}_{M}\left\lbrack x\right\rbrack \), we have thus \( p\left( x\right) = \mathop{\prod }\limits_{{i = 0}}^{{d - 1}}(x - \left. {{\sigma }_{i}\left( \theta \right) }\right) \). For a prime ideal \( \mathfrak{P} \mid \mathfrak{p} \) of \( ... | Yes |
Corollary 1.9.7. For \( L/K \), there are only finitely many \( \mathfrak{p} \subset {\mathcal{O}}_{K} \) that are ramified in \( {\mathcal{O}}_{L} \) . | Proof. Let \( \theta \in {\mathcal{O}}_{L} \) be as above (with \( p\left( x\right) \) the minimal polynomial of \( \theta \) over \( K \), and \( {\mathcal{F}}_{\theta } \) the conduction of \( {\mathcal{O}}_{K}\left\lbrack \theta \right\rbrack \) ). We only need to there are only finitely many \( \mathfrak{p} \subset... | No |
Lemma 1.10.1. Let \( \mathfrak{P} \) be a prime ideal of \( {\mathcal{O}}_{L} \), then \( \sigma \left( \mathfrak{P}\right) \) is also a prime ideal of \( {\mathcal{O}}_{L} \). | Proof. Suppose \( \mathfrak{a}\mathfrak{b} \subset \sigma \left( {fP}\right) \). Then \( {\sigma }^{-1}\left( \mathfrak{a}\right) {\sigma }^{-1}\left( \mathfrak{b}\right) \subset \mathfrak{P} \). Since \( \mathfrak{P} \) is prime, we have \( {\sigma }^{-1}\left( \mathfrak{a}\right) \subset \mathfrak{P} \) or \( {\sigma... | Yes |
Proposition 1.10.2. Let \( \mathfrak{p} \) be a non-zero prime ideal of \( {\mathcal{O}}_{K} \), then \( \operatorname{Gal}\left( {L/K}\right) \) acts transitively on \( \left\{ {{\mathfrak{P}}_{i} \subset {\mathcal{O}}_{L}\left| {\mathfrak{P}}_{i}\right| \mathfrak{p}}\right\} \) . Moreover, \( e\left( {{\mathfrak{P}}_... | Proof. Let \( \mathfrak{P} \mid \mathfrak{p} \), and suppose there exists \( {\mathfrak{P}}_{i} \mid \mathfrak{p} \) such that for all \( \sigma \in \operatorname{Gal}\left( {L/K}\right) ,\sigma \left( \mathfrak{P}\right) \neq {\mathfrak{P}}_{i} \) . So \( {\mathfrak{P}}_{i} \) and \( \{ \sigma \left( \mathfrak{P}\righ... | Yes |
Proposition 1.10.4. (1) The ideal \( \mathfrak{P} \) is the unique prime ideal of \( {\mathcal{O}}_{L} \) such that \( \mathfrak{P} \mid {\mathfrak{P}}_{D} \) . (2) We have \( e\left( {\mathfrak{P}/{\mathfrak{P}}_{D}}\right) = e\left( {\mathfrak{P}/\mathfrak{p}}\right) \left( {\mathfrak{p} = \mathfrak{P} \cap {\mathcal... | Proof. (1) We know \( {D}_{\mathfrak{P}} = \operatorname{Gal}\left( {L/{H}_{\mathfrak{P}}}\right) \) acts transitively on the prime ideals of \( {\mathcal{O}}_{L} \) above \( {\mathfrak{P}}_{D} \) . However, \( \sigma \left( \mathfrak{P}\right) = \mathfrak{P} \) for all \( \sigma \in \operatorname{Gal}\left( {L/{H}_{\m... | Yes |
Proposition 1.10.5. The morphism \( {D}_{\mathfrak{P}}/{I}_{\mathfrak{P}} \rightarrow \operatorname{Gal}\left( {{k}_{\mathfrak{P}}/{k}_{\mathfrak{p}}}\right) \) is an isomorphism. | Proof. First by replacing \( K \) by \( {H}_{\mathfrak{P}} \), we can reduce to the case where \( {D}_{\mathfrak{P}} = \operatorname{Gal}\left( {L/K}\right) \). The injectivity is by definition. Let \( \alpha \in {\mathcal{O}}_{L} \) such that the reduction \( \bar{\alpha } \in {k}_{\mathfrak{P}} \) satisfies \( {k}_{\... | Yes |
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