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Theorem 3.7.1. Let \( {F}^{\prime }/{K}^{\prime } \) be a Galois extension of \( F/K \) and \( {P}_{1},{P}_{2} \in {\mathbb{P}}_{{F}^{\prime }} \) be extensions of \( P \in {\mathbb{P}}_{F} \) . Then \( {P}_{2} = \sigma \left( {P}_{1}\right) \) for some \( \sigma \in \operatorname{Gal}\left( {{F}^{\prime }/F}\right) \)... | Proof. Assume that the assertion is false; i.e., \( \sigma \left( {P}_{1}\right) \neq {P}_{2} \) for all \( \sigma \in G \mathrel{\text{:=}} \) \( \operatorname{Gal}\left( {{F}^{\prime }/F}\right) \) . By the Approximation Theorem there is an element \( z \in {F}^{\prime } \) such that \( {v}_{{P}_{2}}\left( z\right) >... | Yes |
Corollary 3.7.2. Notation as in Theorem 3.7.1 (in particular \( {F}^{\prime }/F \) is a Galois extension). Let \( {P}_{1},\ldots ,{P}_{r} \) be all the places of \( {F}^{\prime } \) lying over \( P \) . Then we have:\n\n(a) \( e\left( {{P}_{i} \mid P}\right) = e\left( {{P}_{j} \mid P}\right) \) and \( f\left( {{P}_{i} ... | Proof. (a) is obvious by Theorem 3.7.1 and Lemma 3.5.2, and (b) is an immediate consequence of (a) and Theorem 3.1.11. As to (c), we have to consider the integral closure\n\n\[ {\mathcal{O}}_{P}^{\prime } = \mathop{\bigcap }\limits_{{i = 1}}^{r}{\mathcal{O}}_{{P}_{i}} \]\n\nof \( {\mathcal{O}}_{P} \) in \( {F}^{\prime ... | Yes |
Proposition 3.7.3 (Kummer Extensions). Let \( F/K \) be an algebraic function field where \( K \) contains a primitive \( n \) -th root of unity (with \( n > 1 \) and \( n \) relatively prime to the characteristic of \( K \) ). Suppose that \( u \in F \) is an element satisfying\n\n\[ u \neq {w}^{d}\;\text{ for all }w ... | We note that every cyclic field extension \( {F}^{\prime }/F \) of degree \( n \) is a Kummer extension, provided that \( n \) is relatively prime to the characteristic of \( F \) and \( F \) contains all \( n \) -th roots of unity. This fact is well-known from Galois theory, cf. Appendix A. | No |
Assume char \( K \neq 2 \) . Let \( F = K\left( {x, y}\right) \) with\n\n\[ \n{y}^{2} = f\left( x\right) = {p}_{1}\left( x\right) \cdot \ldots \cdot {p}_{s}\left( x\right) \in K\left\lbrack x\right\rbrack ,\n\]\n\nwhere \( {p}_{1}\left( x\right) ,\ldots ,{p}_{s}\left( x\right) \) are distinct irreducible monic polynomi... | Proof. We have \( F = {F}_{0}\left( y\right) \) where \( {F}_{0} = K\left( x\right) \) is the rational function field. Let \( {P}_{i} \in {\mathbb{P}}_{K\left( x\right) } \) denote the zero of \( {p}_{i}\left( x\right) \) and \( {P}_{\infty } \) the pole of \( x \) in \( K\left( x\right) \) . Then \( {v}_{{P}_{i}}\left... | Yes |
Proposition 3.7.8 (Artin-Schreier Extensions). Let \( F/K \) be an algebraic function field of characteristic \( p > 0 \). Suppose that \( u \in F \) is an element which satisfies the following condition:\n\n\[ u \neq {w}^{p} - w\;\text{ for all }w \in F. \]\n\nLet\n\n\[ {F}^{\prime } = F\left( y\right) \;\text{ with }... | Proof. (a) This is well-known from Galois theory, see Appendix A.\n\n(b) and (c) First we consider the case \( {m}_{P} = - 1 \) ; i.e., \( {v}_{P}\left( {u - \left( {{z}^{p} - z}\right) }\right) \geq 0 \) for some \( z \in F \). Let \( {y}_{1} = y - z \) and \( {u}_{1} = u - \left( {{z}^{p} - z}\right) \) ; then \( {F}... | Yes |
Theorem 3.8.2. With notation as above the following hold:\n\n(a) The decomposition group \( {G}_{Z}\left( {{P}^{\prime } \mid P}\right) \) has order \( e\left( {{P}^{\prime } \mid P}\right) \cdot f\left( {{P}^{\prime } \mid P}\right) \) . | Proof. (a) By Theorem 3.7.1 \( G \) acts transitively on the set of extensions of \( P \) in \( {F}^{\prime } \) . So we can choose \( {\sigma }_{1},\ldots ,{\sigma }_{r} \in G \) such that \( {\sigma }_{1}\left( {P}^{\prime }\right) ,\ldots ,{\sigma }_{r}\left( {P}^{\prime }\right) \) are all places of \( {F}^{\prime ... | Yes |
Theorem 3.8.3. Consider a Galois extension \( {F}^{\prime }/F \) of algebraic function fields, a place \( P \in {\mathbb{P}}_{F} \) and an extension \( {P}^{\prime } \) of \( F \) in \( {F}^{\prime } \) . For an intermediate field \( F \subseteq M \subseteq {F}^{\prime } \) let \( {P}_{M} \mathrel{\text{:=}} {P}^{\prim... | Proof. By Theorem 3.8.2(d) all implications \( \Rightarrow \) are obvious. Before proving the converse, we remark that the decomposition group of \( {P}^{\prime } \) over \( {P}_{M} \) is contained in \( {G}_{Z}\left( {{P}^{\prime } \mid P}\right) \), and the inertia group of \( {P}^{\prime } \) over \( {P}_{M} \) is c... | Yes |
Proposition 3.8.5. With the above notations we have:\n\n(a) \( {G}_{-1} = {G}_{Z}\left( {{P}^{\prime } \mid P}\right) \) and \( {G}_{0} = {G}_{T}\left( {{P}^{\prime } \mid P}\right) \) . In particular, \( \operatorname{ord}{G}_{0} = e\left( {{P}^{\prime } \mid P}\right) \) .\n\n(b) \( {G}_{-1} \supseteq {G}_{0} \supset... | Proof. (a) and (b) are obvious.\n\n(c) Consider the inertia field \( T \) of \( {P}^{\prime } \) over \( P \), the restriction \( {P}_{T} = {P}^{\prime } \cap T \) and the corresponding valuation ring \( {\mathcal{O}}_{{P}_{T}} = {\mathcal{O}}_{{P}^{\prime }} \cap T \) . The elements \( 1, t,\ldots ,{t}^{e - 1} \) (whe... | Yes |
Theorem 3.8.7 (Hilbert's Different Formula). Consider a Galois extension \( {F}^{\prime }/F \) of algebraic function fields, a place \( P \in {\mathbb{P}}_{F} \) and a place \( {P}^{\prime } \in {\mathbb{P}}_{{F}^{\prime }} \) lying over \( P \) . Then the different exponent \( d\left( {{P}^{\prime } \mid P}\right) \) ... | Proof. First we assume that \( {P}^{\prime } \mid P \) is totally ramified; i.e., \( G \mathrel{\text{:=}} \operatorname{Gal}\left( {{F}^{\prime }/F}\right) = \) \( {G}_{0}\left( {{P}^{\prime } \mid P}\right) \) . Set \( {e}_{i} \mathrel{\text{:=}} \operatorname{ord}{G}_{i}\left( {{P}^{\prime } \mid P}\right) \) (for \... | Yes |
Lemma 3.9.2. Let \( G \) be a finite group and \( U \subseteq G \) be a normal subgroup such that \( \operatorname{ord}U = {p}^{n} \) (with either \( p = 1 \) or else, \( p \) a prime number) and \( G/U \) is cyclic of order relatively prime to p. Suppose that \( {H}_{1} \) is a subgroup of \( G \) with \( {p}^{n} \mid... | Proof of the Lemma. Clearly the order of \( {H}_{1} \cap {H}_{2} \) divides the orders of \( {H}_{1} \) and of \( {H}_{2} \), thus\n\n\[ \operatorname{ord}\left( {{H}_{1} \cap {H}_{2}}\right) \mid \gcd \left( {\operatorname{ord}{H}_{1},\operatorname{ord}{H}_{2}}\right) . \]\n\nWe set ord \( {H}_{1} = {a}_{1}{p}^{n} \) ... | Yes |
Corollary 3.9.3. Let \( {F}^{\prime }/F \) be a finite separable extension of function fields and let \( P \) be a place of \( F \). (a) Suppose that \( {F}^{\prime } = {F}_{1}{F}_{2} \) is the compositum of two intermediate fields \( F \subseteq {F}_{1},{F}_{2} \subseteq {F}^{\prime } \). If \( P \) is unramified in \... | Proof. (a) This is just a special case of Theorem 3.9.1.\n\n(b) The Galois closure \( {F}^{\prime } \) of \( {F}_{0}/F \) is the compositum of the fields \( \sigma \left( {F}_{0}\right) \), where \( \sigma \) runs through all embeddings \( \sigma : {F}_{0} \rightarrow \bar{F} \) over \( F \) (where \( \bar{F} \supseteq... | Yes |
Lemma 3.9.5. Let \( {F}_{0}/F \) be a finite separable extension of function fields and let \( {F}^{\prime } \supseteq {F}_{0} \) be the Galois closure of \( {F}_{0}/F \) . Assume that a place \( P \in {\mathbb{P}}_{F} \) is completely splitting in \( {F}_{0}/F \) . Then \( P \) splits completely in \( {F}^{\prime }/F ... | Proof. Let \( {P}^{\prime } \) be a place of \( {F}^{\prime } \) lying above \( P \) and consider the decomposition field \( Z \mathrel{\text{:=}} Z\left( {{P}^{\prime } \mid P}\right) \subseteq {F}^{\prime } \) (see Definition 3.8.1). Set \( {P}_{0} \mathrel{\text{:=}} {P}^{\prime } \cap {F}_{0} \) . Since \( P \) spl... | Yes |
Proposition 3.9.6. Let \( {F}^{\prime }/F \) be a finite separable extension of function fields and let \( {F}_{1},{F}_{2} \) be intermediate fields of \( {F}^{\prime }/F \) such that \( {F}^{\prime } = {F}_{1}{F}_{2} \) is their compositum.\n\n(a) Suppose that \( P \) is a place of \( F \) which splits completely in t... | Proof. (a) Let \( E/F \) be the Galois closure of \( {F}_{1}/F \) ; then \( P \) splits completely in the extension \( E/F \) by Lemma 3.9.5. We consider the compositum \( {E}^{\prime } \mathrel{\text{:=}} E{F}_{2} \) . By Galois theory we know that the extension \( {E}^{\prime }/{F}_{2} \) is Galois, and the Galois gr... | Yes |
Corollary 3.9.7. Let \( F/K \) be a function field whose full constant field is \( K \) . (a) Suppose that \( {F}^{\prime } = {F}_{1}{F}_{2} \) is the compositum of two finite separable extensions \( {F}_{1}/F \) and \( {F}_{2}/F \) . Assume that there exists a place \( P \in {\mathbb{P}}_{F} \) of degree one which spl... | Proof. (a) We only have to show that \( K \) is the full constant field of \( {F}^{\prime } = {F}_{1}{F}_{2} \) ; the remaining assertions follow immediately from Proposition 3.9.6. We choose a place \( {P}^{\prime } \) of \( {F}^{\prime } \) lying above \( P \), then \( f\left( {{P}^{\prime } \mid P}\right) = 1 \) and... | Yes |
Lemma 3.10.1. Suppose \( {F}^{\prime }/F \) is a purely inseparable field extension of degree \( p \) . Then \( K \) is the constant field of \( {F}^{\prime } \) as well. Every place \( P \in {\mathbb{P}}_{F} \) has only one extension \( {P}^{\prime } \in {\mathbb{P}}_{{F}^{\prime }} \), namely\n\n\[ \n{P}^{\prime } = ... | Proof. Let \( a \in {F}^{\prime } \) be algebraic over \( K \) . Since \( {F}^{\prime }/F \) is purely inseparable of degree \( p \), we have \( {a}^{p} \in F \) and \( {a}^{p} \) is algebraic over \( K \) . As \( K \) is the constant field of \( F \) this shows that \( {a}^{p} \in K \) . But \( K \) is perfect, so \( ... | Yes |
Proposition 3.10.2. (a) Assume \( z \in F \) satisfies \( {v}_{P}\left( z\right) ≢ 0{\;\operatorname{mod}\;p} \) for some \( P \in {\mathbb{P}}_{F} \) . Then \( z \) is a separating element for \( F/K \) . In particular \( F/K \) is separably generated. | Proof. (a) Suppose that \( z \) is not separating. The extension \( F/K\left( z\right) \) is of finite degree since \( z \notin K \), hence there is an intermediate field \( K\left( z\right) \subseteq {F}_{s} \subseteq F \) such that \( F/{F}_{s} \) is purely inseparable of degree \( p \) . Let \( {P}_{s} \mathrel{\tex... | Yes |
Proposition 3.11.1. Let \( {F}_{1}/K \) be a subfield of \( F/K \) and \( \left\lbrack {F : {F}_{1}}\right\rbrack = n \) . Assume that \( \left\{ {{z}_{1},\ldots ,{z}_{n}}\right\} \) is a basis of \( F/{F}_{1} \) such that all \( {z}_{i} \in \mathcal{L}\left( C\right) \) for some divisor \( C \in \operatorname{Div}\lef... | Proof. Let \( {A}_{1} \) be a divisor of \( {F}_{1}/K \) of sufficiently large degree such that\n\n\[ \ell \left( {A}_{1}\right) = : t = \deg {A}_{1} + 1 - {g}_{1}. \]\n\nChoose a basis \( \left\{ {{x}_{1},\ldots ,{x}_{t}}\right\} \subseteq {F}_{1} \) of \( \mathcal{L}\left( {A}_{1}\right) \) . Set \( A \mathrel{\text{... | Yes |
Lemma 3.11.2. Assume that \( K \) is algebraically closed, and consider a subfield \( {F}_{1}/K \) of \( F/K \) such that \( F/{F}_{1} \) is separable of degree \( \left\lbrack {F : {F}_{1}}\right\rbrack = n > 1 \) . Let \( y \in F \) be an element with \( F = {F}_{1}\left( y\right) \) . Then almost all \( P \in {\math... | Proof. Let \( \varphi \left( T\right) = {T}^{n} + {z}_{n - 1}{T}^{n - 1} + \cdots + {z}_{0} \in {F}_{1}\left\lbrack T\right\rbrack \) be the minimal polynomial of \( y \) over \( {F}_{1} \) . For almost all \( P \in {\mathbb{P}}_{{F}_{1}} \) the following hold:\n\n\[ \left\{ {1, y,\ldots ,{y}^{n - 1}}\right\} \text{is ... | Yes |
Proposition 3.11.5. Consider an algebraic function field \( F = K\left( {x, y}\right) \) over \( K \), where the irreducible equation of \( y \) over \( K\left( x\right) \) has the form\n\n\[ \n{y}^{n} + {f}_{1}\left( x\right) {y}^{n - 1} + \ldots + {f}_{n - 1}\left( x\right) y + {f}_{n}\left( x\right) = 0 \n\]\n\n(3.1... | Proof. The proof is similar to that of Proposition 3.11.1. Let \( A \mathrel{\text{:=}} {\left( x\right) }_{\infty } \) denote the pole divisor of \( x \) in \( F \) . It is a positive divisor of degree \( n \) . We claim that\n\n\[ \n{v}_{P}\left( y\right) \geq - {v}_{P}\left( A\right) \text{ for all }P \in {\mathbb{P... | Yes |
Lemma 4.1.2. Let \( \delta : F \rightarrow M \) be a derivation of \( F/K \) into \( M \) . Then we have:\n\n(a) \( \delta \left( a\right) = 0 \) for each \( a \in K \) .\n\n(b) \( \delta \left( {z}^{n}\right) = n{z}^{n - 1} \cdot \delta \left( z\right) \) for \( z \in F \) and \( n \geq 0 \) .\n\n(c) If \( \operatorna... | The simple proof of this lemma can be omitted. | No |
Lemma 4.1.3. Suppose that \( x \) is a separating element of \( F/K \) and that \( {\delta }_{1},{\delta }_{2} \) : \( F \rightarrow M \) are derivations of \( F/K \) with \( {\delta }_{1}\left( x\right) = {\delta }_{2}\left( x\right) \) . Then \( {\delta }_{1} = {\delta }_{2} \) . | Proof. Lemma 4.1.2(b) implies for a polynomial \( f\left( x\right) = \sum {a}_{i}{x}^{i} \in K\left\lbrack x\right\rbrack \) that \( {\delta }_{j}\left( {f\left( x\right) }\right) = \left( {\sum i{a}_{i}{x}^{i - 1}}\right) \cdot {\delta }_{j}\left( x\right) \) for \( j = 1,2 \), hence \( {\delta }_{1}\left( {f\left( x\... | Yes |
Lemma 4.1.6. Let \( x \) be a separating element of \( F/K \) . Then the following hold:\n\n(a) For each derivation \( \eta \in {\operatorname{Der}}_{F} \) we have \( \eta = \eta \left( x\right) \cdot {\delta }_{x} \) . In particular, \( {\operatorname{Der}}_{F} \) is a one-dimensional \( F \) -module.\n\n(b) (Chain ru... | Proof. (a) Consider the two derivations \( \eta \) and \( \eta \left( x\right) \cdot {\delta }_{x} \) of \( F/K \) into \( F \) . Since \( \left( {\eta \left( x\right) \cdot {\delta }_{x}}\right) \left( x\right) = \eta \left( x\right) \cdot {\delta }_{x}\left( x\right) = \eta \left( x\right) \) and \( x \) is separatin... | Yes |
Proposition 4.1.8. (a) Let \( z \in F \) be separating. Then \( {dz} \neq 0 \), and every differential \( \omega \in {\Delta }_{F} \) can uniquely be written in the form \( \omega = {udz} \) with \( u \in F \) . Hence \( {\Delta }_{F} \) is a one-dimensional \( F \) -module. | Proof. (a) The differential \( 0 = {0dz} \) is the zero element of \( {\Delta }_{F} \) . By (4.6) we see immediately that \( \left( {0, z}\right) \) is not equivalent to \( \left( {1, z}\right) \), hence \( {dz} \neq 0 \) .\n\nConsider now an arbitrary differential \( \omega \in {\Delta }_{F} \), say \( \omega = {vdy} ... | Yes |
Lemma 4.2.4. Let \( {\left( {z}_{n}\right) }_{n \geq 0} \) be a sequence in a complete valued field \( \left( {T, v}\right) \) . Then we have: The infinite series \( \mathop{\sum }\limits_{{i = 0}}^{\infty }{z}_{i} \) is convergent if and only if the sequence \( {\left( {z}_{n}\right) }_{n \geq 0} \) converges to 0 . | Proof. Suppose that \( {\left( {z}_{n}\right) }_{n \geq 0} \) converges to 0 . Consider the \( m \) -th partial sum \( {s}_{m} \mathrel{\text{:=}} \mathop{\sum }\limits_{{i = 0}}^{m}{z}_{i} \) . For \( n > m \) we have\n\n\[ v\left( {{s}_{n} - {s}_{m}}\right) = v\left( {\mathop{\sum }\limits_{{i = m + 1}}^{n}{z}_{i}}\r... | Yes |
Proposition 4.2.7. Let \( P \) be a place of \( F/K \) of degree one and let \( t \in F \) be a P-prime element. If \( z \in F \) has the P-adic expansion \( z = \mathop{\sum }\limits_{{i = n}}^{\infty }{a}_{i}{t}^{i} \) with coefficients \( {a}_{i} \in K \), then\n\n\[ \frac{dz}{dt} = \mathop{\sum }\limits_{{i = n}}^{... | Proof. We define a mapping \( \delta : {\widehat{F}}_{P} \rightarrow {\widehat{F}}_{P} \) by\n\n\[ \delta \left( {\mathop{\sum }\limits_{{i = m}}^{\infty }{c}_{i}{t}^{i}}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{i = m}}^{\infty }i{c}_{i}{t}^{i - 1}. \]\n\nThis mapping is obviously \( K \) -linear and satisfie... | Yes |
Proposition 4.2.9. Let \( s, t \in F \) be \( P \) -prime elements (where \( P \) is a place of degree one). Then\n\n\[ \n{\operatorname{res}}_{P, s}\left( z\right) = {\operatorname{res}}_{P, t}\left( {z \cdot \frac{ds}{dt}}\right)\n\]\n\nfor all \( z \in F \) . | Proof. The power series expansion of \( s \) with respect to \( t \) has the following form (see Theorem 4.2.6):\n\n\[ \ns = \mathop{\sum }\limits_{{i = 1}}^{\infty }{c}_{i}{t}^{i}\;\text{ with }\;{c}_{1} \neq 0.\n\]\n\nProposition 4.2.7 yields\n\n\[ \n\frac{ds}{dt} = {c}_{1} + \mathop{\sum }\limits_{{i = 2}}^{\infty }... | Yes |
Corollary 4.3.3 (Residue Theorem). Let \( F/K \) be an algebraic function field over an algebraically closed field, and let \( \omega \in {\Delta }_{F} \) be a differential of \( F/K \) . Then \( {\operatorname{res}}_{P}\left( \omega \right) = 0 \) for almost all places \( P \in {\mathbb{P}}_{F} \), and\n\n\[ \mathop{\... | Proof of the Corollary. Write \( \omega = {zdx} \) with \( z \in F \) and a separating element \( x \in F \) . By Theorem 4.3.2(d) we have \( {\operatorname{res}}_{P}\left( \omega \right) = {\left( z \cdot \delta \left( x\right) \right) }_{P}\left( 1\right) \) . Now Proposition 1.7.2 yields the desired result. | Yes |
Lemma 4.3.5. Let \( F \) be an algebraic function field over an algebraically closed field \( K \) . Suppose that \( x \) is a separating element of \( F/K \) and \( {P}_{0} \in {\mathbb{P}}_{K\left( x\right) } \) satisfies the following conditions:\n\n(1) \( {P}_{0} \) is unramified in \( F/K\left( x\right) \) .\n\n(2... | Proof. By (1) and (2) there exists an element \( a \in K \) such that \( t \mathrel{\text{:=}} x - a \) is a \( P \) - prime element. As always, \( {P}_{\infty } \) denotes the pole of \( x \) in \( K\left( x\right) \) . Consider the Weil differential \( \eta \in {\Omega }_{K\left( x\right) } \) which is given by (4.22... | Yes |
(c) As an important special case of Theorem 3.4.6 we obtain the following formula for the divisor of a differential \( \omega = {zdx} \neq 0 \) : | \[ \left( {zdx}\right) = \left( z\right) + \left( {dx}\right) = \left( z\right) - 2{\left( x\right) }_{\infty } + \operatorname{Diff}\left( {F/K\left( x\right) }\right) . \] | Yes |
Lemma 5.1.1. For every \( n \geq 0 \) there exist only finitely many positive divisors of degree \( n \) . | Proof. A positive divisor is a sum of prime divisors. Hence it is sufficient to prove that the set \( S \mathrel{\text{:=}} \left\{ {P \in {\mathbb{P}}_{F} \mid \deg P \leq n}\right\} \) is finite. We choose an element \( x \in F \smallsetminus {\mathbb{F}}_{q} \) and consider the set \( {S}_{0} \mathrel{\text{:=}} \le... | Yes |
Proposition 5.1.3. \( {\mathrm{{Cl}}}^{0}\left( F\right) \) is a finite group. Its order \( h = {h}_{F} \) is called the class number of \( F/{\mathbb{F}}_{q} \) ; i.e., \[ h \mathrel{\text{:=}} {h}_{F} \mathrel{\text{:=}} \operatorname{ord}{\operatorname{Cl}}^{0}\left( F\right) . \] | Proof. Choose a divisor \( B \in \operatorname{Div}\left( F\right) \) of degree \( \geq g \), say \( n \mathrel{\text{:=}} \deg B \), and consider the set of divisor classes \[ {\mathrm{{Cl}}}^{n}\left( F\right) \mathrel{\text{:=}} \{ \left\lbrack C\right\rbrack \in \mathrm{{Cl}}\left( F\right) \mid \deg \left\lbrack C... | Yes |
Lemma 5.1.4. (a) \( {A}_{n} = 0 \) if \( \partial \nmid n \) . | Proof. (a) is trivial. | No |
Proposition 5.1.6. The power series \( Z\left( t\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{A}_{n}{t}^{n} \) is convergent for \( \left| t\right| < {q}^{-1} \) . More precisely, we have for \( \left| t\right| < {q}^{-1} \) : (a) If \( F/{\mathbb{F}}_{q} \) has genus \( g = 0 \) then\n\n\[ Z\left( t\right) = \fr... | Proof. (a) \( g = 0 \) . To begin with, we show that a function field of genus zero has class number \( h = 1 \) ; i.e., every divisor \( A \) of degree 0 is principal. This fact follows easily from the Riemann-Roch Theorem: as \( 0 > {2g} - 2 \), we have \( \ell \left( A\right) = \deg A + 1 - g = 1 \), and we can ther... | Yes |
Corollary 5.1.7. \( Z\\left( t\\right) \) can be extended to a rational function on \( \\mathbb{C} \) ; it has a simple pole at \( t = 1 \) . | Proof. Obvious, since \( 1/\\left( {1 - {t}^{\\partial }}\\right) \) has a simple pole at \( t = 1 \) . | Yes |
Proposition 5.1.8 (Euler Product). For \( \left| t\right| < {q}^{-1} \) the Zeta function can be represented as an absolutely convergent product\n\n\[ Z\left( t\right) = \mathop{\prod }\limits_{{P \in {\mathbb{P}}_{F}}}{\left( 1 - {t}^{\deg P}\right) }^{-1}. \] | Proof. The right hand side of (5.5) converges absolutely for \( \left| t\right| < {q}^{-1} \), since \( \mathop{\sum }\limits_{{P \in {\mathbb{P}}_{F}}}{\left| t\right| }^{\deg P} \leq \mathop{\sum }\limits_{{n = 0}}^{\infty }{A}_{n}{\left| t\right| }^{n} < \infty \) by Proposition 5.1.6. Each factor of (5.5) can be wr... | Yes |
Lemma 5.1.9. (a) \( {F}_{r}/F \) is a cyclic extension of degree \( r \) (i.e., \( {F}_{r}/F \) is Galois with a cyclic Galois group of order \( r \) ). The Galois group \( \operatorname{Gal}\left( {{F}_{r}/F}\right) \) is generated by the Frobenius automorphism \( \sigma \) which acts on \( {\mathbb{F}}_{{q}^{r}} \) b... | Proof. (a) It is well-known that \( {\mathbb{F}}_{{q}^{r}}/{\mathbb{F}}_{q} \) is cyclic of degree \( r \), and its Galois group is generated by the Frobenius map \( \alpha \mapsto {\alpha }^{q} \) . Since \( \left\lbrack {{F}_{r} : F}\right\rbrack = \left\lbrack {{\mathbb{F}}_{{q}^{r}} : {\mathbb{F}}_{q}}\right\rbrack... | Yes |
Proposition 5.1.10. Let \( Z\left( t\right) \) (resp. \( {Z}_{r}\left( t\right) \) ) denote the Zeta function of \( F \) (resp. of \( {F}_{r} = F{\mathbb{F}}_{{q}^{r}} \) ). Then\n\n\[ \n{Z}_{r}\left( {t}^{r}\right) = \mathop{\prod }\limits_{{{\zeta }^{r} = 1}}Z\left( {\zeta t}\right) \n\]\n\n(5.8)\n\nfor all \( t \in ... | Proof. It is sufficient to prove (5.8) for \( \left| t\right| < {q}^{-1} \) . In this region the Euler product representation yields\n\n\[ \n{Z}_{r}\left( {t}^{r}\right) = \mathop{\prod }\limits_{{P \in {\mathbb{P}}_{F}}}\mathop{\prod }\limits_{{{P}^{\prime } \mid P}}{\left( 1 - {t}^{r \cdot \deg {P}^{\prime }}\right) ... | Yes |
Corollary 5.1.11 (F.K. Schmidt). \( \partial = 1 \) . | Proof. For \( {\zeta }^{\partial } = 1 \) we have\n\n\[ Z\left( {\zeta t}\right) = \mathop{\prod }\limits_{{P \in {\mathbb{P}}_{F}}}{\left( 1 - {\left( \zeta t\right) }^{\deg P}\right) }^{-1} = \mathop{\prod }\limits_{{P \in {\mathbb{P}}_{F}}}{\left( 1 - {t}^{\deg P}\right) }^{-1} = Z\left( t\right) ,\]\n\nsince \( \pa... | Yes |
Corollary 5.1.12. (a) Every function field \( F/{\mathbb{F}}_{q} \) of genus 0 is rational, and its Zeta function is\n\n\[ Z\left( t\right) = \frac{1}{\left( {1 - t}\right) \left( {1 - {qt}}\right) }.\n\] | Proof. A function field of genus 0 having a divisor of degree 1 is rational, cf. Proposition 1.6.3. The remaining assertions follow from Proposition 5.1.6 and \( \partial = 1 \) . | No |
Proposition 5.1.13 (Functional Equation of the Zeta Function). The Zeta function of \( F/{\mathbb{F}}_{q} \) satisfies the functional equation\n\n\[ Z\left( t\right) = {q}^{g - 1}{t}^{{2g} - 2}Z\left( \frac{1}{qt}\right) . \] | Proof. For \( g = 0 \) this is obvious from Corollary 5.1.12(a). For \( g \geq 1 \) we write \( Z\left( t\right) = F\left( t\right) + G\left( t\right) \) as in Corollary 5.1.12(b). Let \( W \) be a canonical divisor of \( F \) ; then\n\n\[ \left( {q - 1}\right) F\left( t\right) = \mathop{\sum }\limits_{{0 \leq \deg \le... | Yes |
Theorem 5.1.15. (a) \( L\left( t\right) \in \mathbb{Z}\left\lbrack t\right\rbrack \) and \( \deg L\left( t\right) = {2g} \) . | All assertions are trivial for \( g = 0 \), hence we can assume from now on that \( g \geq 1 \) . (a) We have already remarked that \( L\left( t\right) \) is a polynomial of degree \( \leq {2g} \) . In (d) we shall prove that its leading coefficient is \( {q}^{g} \), so \( \deg L\left( t\right) = {2g} \) . The assertio... | No |
Corollary 5.1.16. For all \( r \geq 1 \) ,\n\n\[ \n{N}_{r} = {q}^{r} + 1 - \mathop{\sum }\limits_{{i = 1}}^{{2g}}{\alpha }_{i}^{r} \n\]\n\nwhere \( {\alpha }_{1},\ldots ,{\alpha }_{2g} \in \mathbb{C} \) are the reciprocals of the roots of \( L\left( t\right) \) . In particular, since \( {N}_{1} = N\left( F\right) \), w... | Proof. By Theorem 5.1.15(d), \( {N}_{r} - \left( {{q}^{r} + 1}\right) \) is the coefficient of \( t \) in the \( L \) - polynomial \( {L}_{r}\left( t\right) \) . On the other hand, since \( {L}_{r}\left( t\right) = \mathop{\prod }\limits_{{i = 1}}^{{2g}}\left( {1 - {\alpha }_{i}^{r}t}\right) \), this coefficient is \( ... | Yes |
Corollary 5.1.17. Let \( L\left( t\right) = \mathop{\sum }\limits_{{i = 0}}^{{2g}}{a}_{i}{t}^{i} \) be the L-polynomial of \( F/{\mathbb{F}}_{q} \), and \( {S}_{r} \mathrel{\text{:=}} {N}_{r} - \left( {{q}^{r} + 1}\right) \) . Then we have:\n\n(a) \( {L}^{\prime }\left( t\right) /L\left( t\right) = \mathop{\sum }\limit... | Proof. (a) Write \( L\left( t\right) = \mathop{\prod }\limits_{{i = 1}}^{{2g}}\left( {1 - {\alpha }_{i}t}\right) \) as in (5.13). Then\n\n\[ {L}^{\prime }\left( t\right) /L\left( t\right) = \mathop{\sum }\limits_{{i = 1}}^{{2g}}\frac{-{\alpha }_{i}}{\left( 1 - {\alpha }_{i}t\right) } = \mathop{\sum }\limits_{{i = 1}}^{... | Yes |
Theorem 5.2.3 (Hasse-Weil Bound). The number \( N = N\left( F\right) \) of places of \( F/{\mathbb{F}}_{q} \) of degree one satisfies the inequality\n\n\[ \left| {N - \left( {q + 1}\right) }\right| \leq {2g}{q}^{1/2} \] | Proof. Corollary 5.1.16 yields\n\n\[ N - \left( {q + 1}\right) = - \mathop{\sum }\limits_{{i = 1}}^{{2g}}{\alpha }_{i} \]\n\nHence the Hasse-Weil Bound is an immediate consequence of the Hasse-Weil Theorem. | No |
Lemma 5.2.4. Let \( m \geq 1 \) . Then the Hasse-Weil Theorem holds for \( F/{\mathbb{F}}_{q} \) if and only if it holds for the constant field extension \( {F}_{m}/{\mathbb{F}}_{{q}^{m}} \) . | Proof. The reciprocals of the roots of \( {L}_{F}\left( t\right) \) are \( {\alpha }_{1},\ldots ,{\alpha }_{2g} \) . By Theorem 5.1.15(f), the reciprocals of the roots of \( {L}_{m}\left( t\right) \) are \( {\alpha }_{1}^{m},\ldots ,{\alpha }_{2g}^{m} \) (as in Theorem 5.1.15, we denote by \( {L}_{m}\left( t\right) \) ... | Yes |
Lemma 5.2.5. Assume there is a constant \( c \in \mathbb{R} \) such that for all \( r \geq 1 \) ,\n\n\[ \left| {{N}_{r} - \left( {{q}^{r} + 1}\right) }\right| \leq c{q}^{r/2} \]\n\nThen the Hasse-Weil Theorem holds for \( F/{\mathbb{F}}_{q} \) . | Proof. Corollary 5.1.16 states that \( {N}_{r} - \left( {{q}^{r} + 1}\right) = - \mathop{\sum }\limits_{{i = 1}}^{{2g}}{\alpha }_{i}^{r} \), hence (5. yields\n\n\[ \left| {\mathop{\sum }\limits_{{i = 1}}^{{2g}}{\alpha }_{i}^{r}}\right| \leq c{q}^{r/2} \]\n\n(5.21)\n\nfor all \( r \geq 1 \) . Consider the meromorphic fu... | Yes |
Proposition 5.2.6. Suppose that \( F/{\mathbb{F}}_{q} \) satisfies the following assumptions:\n\n\[ \text{(1)}q\text{is a square, and (2)}q > {\left( g + 1\right) }^{4}\text{.} \]\n\nThen the number \( N = N\left( F\right) \) of places of \( F/{\mathbb{F}}_{q} \) of degree one can be estimated \( {by} \)\n\n\[ N < \lef... | Proof. We can assume that there exists a place \( Q \in {\mathbb{P}}_{F} \) of degree one (otherwise \( N = 0 \), and the proposition is trivial). Set\n\n\[ {q}_{0} \mathrel{\text{:=}} {q}^{1/2}, m \mathrel{\text{:=}} {q}_{0} - 1\;\text{ and }\;n \mathrel{\text{:=}} {2g} + {q}_{0}. \]\n\nOne checks easily that\n\n\[ r ... | Yes |
Lemma 5.2.7. Let \( {G}^{\prime } \) be a group which is the direct product\n\n\[ \n{G}^{\prime } = \langle \sigma \rangle \times G \n\]\n\n(5.31)\n\nof a cyclic subgroup \( \langle \sigma \rangle \) and a subgroup \( G \subseteq {G}^{\prime } \) such that \( \operatorname{ord}G = m,\operatorname{ord}\left( \sigma \rig... | Proof. For \( \tau \in G \) we consider the cyclic subgroup \( \langle {\sigma \tau }\rangle \subseteq {G}^{\prime } \) . Since \( {\sigma \tau } = {\tau \sigma } \) (by (5.31)), ord \( \left( \sigma \right) = n \) and \( \operatorname{ord}\left( \tau \right) \mid m \) we conclude that \( \operatorname{ord}\left( {\sig... | Yes |
Corollary 5.2.10. (a) The estimate\n\n\[ \n\\left| {{B}_{r} - \\frac{{q}^{r}}{r}}\\right| \\leq \\left( {\\frac{q}{q - 1} + {2g}\\frac{{q}^{1/2}}{{q}^{1/2} - 1}}\\right) \\cdot \\frac{{q}^{r/2} - 1}{r} < \\left( {2 + {7g}}\\right) \\cdot \\frac{{q}^{r/2}}{r} \n\]\n\nholds for all \( r \\geq 1 \) . | Proof. (a) For \( r = 1 \) we have \( {B}_{1} = N \), and the assertion follows easily from the Hasse-Weil Bound. For \( r \\geq 2 \) Proposition 5.2.9 yields\n\n\[ \n{B}_{r} - \\frac{{q}^{r}}{r} = \\frac{1}{r}\\mathop{\\sum }\\limits_{{d \\mid r, d < r}}\\mu \\left( \\frac{r}{d}\\right) {q}^{d} + \\frac{1}{r}\\mathop{... | Yes |
Proposition 5.3.3 (Ihara). Suppose that \( F/{\mathbb{F}}_{q} \) is a maximal function field. Then \( g \leq \left( {q - {q}^{1/2}}\right) /2 \) . | Proof. Let \( {\alpha }_{1},\ldots ,{\alpha }_{2g} \) be the reciprocals of the roots of \( L\left( t\right) \) . Since\n\n\[ N = q + 1 - \mathop{\sum }\limits_{{i = 1}}^{{2g}}{\alpha }_{i}\;\text{ and }\;\left| {\alpha }_{i}\right| = {q}^{1/2} \]\n\n(by Corollary 5.1.16 and Theorem 5.2.1), the assumption \( N = q + 1 ... | Yes |
Proposition 5.3.4 (Serre’s Explicit Formulas). Suppose that \( {c}_{1},\ldots ,{c}_{m} \in \) \( \mathbb{R} \) satisfy the following conditions:\n\n(1) \( {c}_{r} \geq 0 \) for \( r = 1,\ldots, m \), and not all \( {c}_{r} = 0 \).\n\n(2) \( {f}_{m}\left( t\right) \geq 0 \) for all \( t \in \mathbb{C} \) with \( \left| ... | Proof. We have \( N = {N}_{1} \leq {N}_{r} \) for all \( r \geq 1 \) . So (5.54) and the assumptions (1) and (2) imply\n\n\[ N \cdot {\lambda }_{m}\left( {q}^{-1/2}\right) \leq {\lambda }_{m}\left( {q}^{1/2}\right) + {\lambda }_{m}\left( {q}^{-1/2}\right) + g. \]\n\nDividing this inequality by \( {\lambda }_{m}\left( {... | Yes |
Proposition 6.1.3. (a) \( \operatorname{char}K \neq 2 \) . Suppose that \( F = K\left( {x, y}\right) \) with\n\n\[ \n{y}^{2} = f\left( x\right) \in K\left\lbrack x\right\rbrack \n\]\n\nwhere \( f\left( x\right) \) is a square-free polynomial of degree 3 . Consider the decomposition \( f\left( x\right) = c\mathop{\prod ... | Proof. In case char \( K \neq 2 \), all assertions follow easily from Proposition 3.7.3 (see also Corollary 3.7.4 and Example 3.7.6). | Yes |
We would like to investigate some elliptic function fields \( F \) over the field \( {\mathbb{F}}_{2} \). Let \( N \) denote the number of places of \( F/{\mathbb{F}}_{2} \) of degree one. The Serre Bound states that\n\n\[ N \leq 2 + 1 + g \cdot \left\lbrack {2\sqrt{2}}\right\rbrack = 5. \] | Let us show that (up to isomorphism) there exists exactly one elliptic function field \( F/{\mathbb{F}}_{2} \) with \( N = 5 \). By Proposition 6.1.2(b) we can write \( F = {\mathbb{F}}_{2}\left( {x, y}\right) \n\nwith\n\[ {y}^{2} + y = x + \frac{1}{x + b},\;b \in {\mathbb{F}}_{2} \]\n\n(6.8)\n\nor\n\n\[ {y}^{2} + y = ... | Yes |
Proposition 6.1.6. Let \( F/K \) be an elliptic function field. Define\n\n\[ \n{\mathbb{P}}_{F}^{\left( 1\right) } \mathrel{\text{:=}} \left\{ {P \in {\mathbb{P}}_{F} \mid \deg P = 1}\right\} \n\] \n\nThen the following hold:\n\n(a) For each divisor \( A \in \operatorname{Div}\left( F\right) \) with \( \deg A = 1 \) th... | Proof. (a) Let \( A \in \operatorname{Div}\left( F\right) \) and \( \deg A = 1 \) . We show the existence of a place \( P \in {\mathbb{P}}_{F}^{\left( 1\right) } \) with \( A \sim P \) as in the proof of Proposition 6.1.2; since \( \ell \left( A\right) = \) \( \deg A + 1 - g > 0 \), there is a divisor \( {A}_{1} \sim A... | Yes |
Proposition 6.1.7. Let \( F/K \) be an elliptic function field. Then:\n\n(a) \( {\mathbb{P}}_{F}^{\left( 1\right) } \) is an abelian group with respect to the operation \( \oplus \) as defined in (6.14).\n\n(b) The place \( {P}_{0} \) is the zero element of the group \( {\mathbb{P}}_{F}^{\left( 1\right) } \) .\n\n(c) F... | Proof. (a), (b) and (d) are obvious.\n\n(c) By (6.14) we have the following equivalences:\n\n\[ P \oplus Q = R \Leftrightarrow \Phi \left( R\right) = \Phi \left( P\right) + \Phi \left( Q\right) \]\n\n\[ \Leftrightarrow R - {P}_{0} \sim \left( {P - {P}_{0}}\right) + \left( {Q - {P}_{0}}\right) \]\n\n\[ \Leftrightarrow P... | Yes |
Lemma 6.2.2. (a) A function field \( F/K \) of genus \( g \geq 2 \) is hyperelliptic if and only if there exists a divisor \( A \in \operatorname{Div}\left( F\right) \) with \( \deg A = 2 \) and \( \ell \left( A\right) \geq 2 \) . | Proof. (a) Suppose that \( F/K \) is hyperelliptic. Choose an element \( x \in F \) such that \( \left\lbrack {F : K\left( x\right) }\right\rbrack = 2 \), and consider the divisor \( A \mathrel{\text{:=}} {\left( x\right) }_{\infty } \) . Then \( \deg A = 2 \) and the elements \( 1, x \in \mathcal{L}\left( A\right) \) ... | Yes |
Proposition 6.2.3. Assume that \( \operatorname{char}K \neq 2 \) .\n\n(a) Let \( F/K \) be a hyperelliptic function field of genus \( g \) . Then there exist \( x, y \in F \) such that \( F = K\left( {x, y}\right) \) and\n\n\[ {y}^{2} = f\left( x\right) \in K\left\lbrack x\right\rbrack \]\n\nwith a square-free polynomi... | Proof. (a) As \( F/K\left( x\right) \) is cyclic of degree 2 and \( \operatorname{char}K \neq 2 \), there exists an element \( z \in F \) such that \( F = K\left( {x, z}\right) \) and \( {z}^{2} = u\left( x\right) \in K\left( x\right) \) . Write\n\n\[ u\left( x\right) = c \cdot \prod {p}_{i}{\left( x\right) }^{{r}_{i}}... | Yes |
Proposition 6.2.4. Consider a hyperelliptic function field \( F/K \) of genus \( g \) and a rational subfield \( K\left( x\right) \subseteq F \) with \( \left\lbrack {F : K\left( x\right) }\right\rbrack = 2 \) . Then the following hold:\n\n(a) All rational subfields \( K\left( z\right) \subseteq F \) with \( \left\lbra... | Proof. (a) Suppose that \( \left\lbrack {F : K\left( z\right) }\right\rbrack \leq g \) but \( z \notin K\left( x\right) \) . Then \( F = K\left( {x, z}\right) \) , and Riemann's Inequality (Theorem 3.11.4) yields the contradiction\n\n\[ g \leq \left( {\left\lbrack {F : K\left( x\right) }\right\rbrack - 1}\right) \cdot ... | Yes |
Proposition 6.3.1. Suppose that \( F = K\left( {x, y}\right) \) is defined by (6.18) and (6.19). Then we have:\n\n(a) \( K \) is the full constant field of \( F \), and \( \left\lbrack {F : K\left( x\right) }\right\rbrack = n \) . If \( K \) contains a primitive \( n \) -th root of unity, \( F/K\left( x\right) \) is a ... | Proof. All assertions follow immediately from Proposition 3.7.3, Corollary 3.7.4 and Remark 3.7.5. | No |
The special case \( H \mathrel{\text{:=}} {\mathbb{F}}_{{q}^{2}}\left( {x, y}\right) \) with\n\n\[ \n{x}^{q + 1} + {y}^{q + 1} = 1 \n\]\n\nis called the Hermitian function field over \( {\mathbb{F}}_{{q}^{2}} \) . | It is a maximal function field by (6.23), so it provides an example of a maximal function field of genus \( g = q\left( {q - 1}\right) /2 \) and shows that Proposition 5.3.3 cannot be improved. The number of places of degree one is \( N = 1 + {q}^{3} \) . | No |
We consider the function field \( F = K\left( {y, z}\right) \) defined by\n\n\[ \n{z}^{3} + {y}^{3}z + y = 0.\n\]\n\n\( F \) is called the function field of the Klein Quartic. The polynomial \( {T}^{3} + {y}^{3}T + y \in K\left( y\right) \left\lbrack T\right\rbrack \) is absolutely irreducible (by Proposition 3.1.15), ... | It is convenient to choose other generators of \( F/K \). We multiply (6.29) by \( {y}^{6} \), set \( x \mathrel{\text{:=}} - {y}^{2}z \) and obtain \( F = K\left( {x, y}\right) \) with\n\n\[ \n{y}^{7} = {x}^{3}/\left( {1 - x}\right)\n\]\n\nIf char \( K = 7, F/K\left( x\right) \) is purely inseparable; therefore \( F/K... | Yes |
Proposition 6.4.1. Consider a function field \( F = K\left( {x, y}\right) \) with\n\n\[ \n{y}^{q} + {\mu y} = f\left( x\right) \in K\left\lbrack x\right\rbrack \n\]\n\n(6.32)\n\nwhere \( q = {p}^{s} > 1 \) is a power of \( p \) and \( 0 \neq \mu \in K \) . Assume that \( \deg f = : m > 0 \) is prime to \( p \), and tha... | Proof. Equation (6.32) is a special case of the situation which was considered in Proposition 3.7.10, so (a) - (e) hold.\n\n(g) \( {\left( x\right) }_{\infty } = q{Q}_{\infty } \) follows from (c). The elements \( x \) and \( y \) have the same poles, hence \( {Q}_{\infty } \) is the only pole of \( y \) as well. Since... | Yes |
We consider a special case of the previous proposition, namely\n\n\\[ F = {\\mathbb{F}}_{{q}^{2}}\\left( {x, y}\\right) \\;\\text{ with }\\;{y}^{q} + y = {x}^{m}\\;\\text{ and }\\;m \\mid \\left( {q + 1}\\right) .\n\\]\n\nThe genus of \\( F \\) is \\( g = \\left( {q - 1}\\right) \\left( {m - 1}\\right) /2 \\) . We clai... | The pole \\( {Q}_{\\infty } \\) of \\( x \\) is one of them. The other places of degree one are extensions of some place \\( {P}_{\\alpha } \\in {\\mathbb{P}}_{K\\left( x\\right) } \\) . Hence, by Proposition 6.4.1(i), we have to count the elements \\( \\alpha \\in {\\mathbb{F}}_{{q}^{2}} \\) such that the equation\n\n... | Yes |
The Hermitian function field \( H \) which was studied in Example 6.3.6 is given by \[ H = {\mathbb{F}}_{{q}^{2}}\left( {u, v}\right) \;\text{ with }\;{u}^{q + 1} + {v}^{q + 1} = 1. \] | We choose \( a, b, c \in {\mathbb{F}}_{{q}^{2}} \) such that \[ {a}^{q + 1} = - 1,\;{b}^{q} + b = 1\;\text{ and }\;c = - a{b}^{q}; \] then it follows that \[ a{b}^{q} + c = 0, \] \[ {a}^{q}b + {c}^{q} = {\left( a{b}^{q} + c\right) }^{q} = 0, \] \[ a{c}^{q} + {a}^{q}c = a\left( {-{a}^{q}b}\right) + {a}^{q}\left( {-a{b}^... | Yes |
Lemma 7.2.3. Let \( \mathcal{F} = \left( {{F}_{0},{F}_{1},{F}_{2},\ldots }\right) \) be a tower over \( {\mathbb{F}}_{q} \) . Then the following hold:\n\n(a) The sequence of rational numbers \( {\left( N\left( {F}_{i}\right) /\left\lbrack {F}_{i} : {F}_{0}\right\rbrack \right) }_{i \geq 0} \) is monotonically decreasin... | Proof. (a) If \( Q \) is a rational place of \( {F}_{i + 1} \), then the restriction \( P \mathrel{\text{:=}} Q \cap {F}_{i} \) of \( Q \) to \( {F}_{i} \) is a rational place of \( {F}_{i} \) . Conversely, at most \( \left\lbrack {{F}_{i + 1} : {F}_{i}}\right\rbrack \) rational places of \( {F}_{i + 1} \) lie above a ... | Yes |
Proposition 7.2.8. Let \( \mathcal{E} \) be a subtower of \( \mathcal{F} \) . Then \( \lambda \left( \mathcal{E}\right) \geq \lambda \left( \mathcal{F}\right) \) . In particular one has:\n\n(a) If \( \mathcal{F} \) is asymptotically good then \( \mathcal{E} \) is also asymptotically good.\n\n(b) If \( \mathcal{E} \) is... | Proof. Let \( {\varphi }_{i} : {E}_{i} \rightarrow {F}_{j\left( i\right) } \) be an embedding of \( {E}_{i} \) into \( {F}_{j\left( i\right) } \) . Let \( {H}_{i} \) be the subfield of \( {F}_{j\left( i\right) } \) which is uniquely determined by the following properties:\n\n- \( {\varphi }_{i}\left( {E}_{i}\right) \su... | Yes |
Theorem 7.2.10. Let \( \mathcal{F} = \left( {{F}_{0},{F}_{1},{F}_{2},\ldots }\right) \) be a tower over \( {\mathbb{F}}_{q} \) . (a) Let \( s \mathrel{\text{:=}} \left| {\operatorname{Split}\left( {\mathcal{F}/{F}_{0}}\right) }\right| \) . Then the splitting rate \( \nu \left( {\mathcal{F}/{F}_{0}}\right) \) satisfies ... | Proof. (a) Above each place \( P \in \operatorname{Split}\left( {\mathcal{F}/{F}_{0}}\right) \) there are exactly \( \left\lbrack {{F}_{n} : {F}_{0}}\right\rbrack \) places of \( {F}_{n} \), and they are all rational. Hence \( N\left( {F}_{n}\right) \geq \left\lbrack {{F}_{n} : {F}_{0}}\right\rbrack \cdot \left| {\oper... | Yes |
Corollary 7.2.11. Let \( \mathcal{F} = \left( {{F}_{0},{F}_{1},{F}_{2},\ldots }\right) \) be a tame tower. Assume that the splitting locus \( \operatorname{Split}\left( {\mathcal{F}/{F}_{0}}\right) \) is non-empty and that the ramification locus \( \operatorname{Ram}\left( {\mathcal{F}/{F}_{0}}\right) \) is finite. The... | Proof. For a tamely ramified place \( Q \mid P \) we have \( d\left( {Q \mid P}\right) = e\left( {Q \mid P}\right) - 1 \leq \) \( e\left( {Q \mid P}\right) \), by Dedekind’s Different Theorem. Therefore we can choose \( {a}_{P} \mathrel{\text{:=}} 1 \) in (7.7), and Theorem 7.2.10(c) gives the desired result. | Yes |
Proposition 7.2.15. Consider a sequence of fields \( {F}_{0} \subseteq {F}_{1} \subseteq {F}_{2} \subseteq \ldots \) where \( {F}_{0} \) is a function field with the exact constant field \( {\mathbb{F}}_{q} \) and \( \left\lbrack {{F}_{n + 1} : {F}_{n}}\right\rbrack \) \( < \infty \) for all \( n \geq 0 \) . Suppose th... | Proof. By the Fundamental Equality we have \( \left\lbrack {{F}_{n + 1} : {F}_{n}}\right\rbrack \geq e\left( {{Q}_{n} \mid {P}_{n}}\right) \) and therefore \( {F}_{n} \subsetneqq {F}_{n + 1} \) . If we assume the equality \( e\left( {{Q}_{n} \mid {P}_{n}}\right) = \left\lbrack {{F}_{n + 1} : {F}_{n}}\right\rbrack \), t... | Yes |
Proposition 7.2.20. Let \( \mathcal{F} = \left( {{F}_{0},{F}_{1},{F}_{2},\ldots }\right) \) be a tower over \( {\mathbb{F}}_{q} \) which is recursively defined by the equation \( f\left( Y\right) = h\left( X\right) \), and let \( F = {\mathbb{F}}_{q}\left( {x, y}\right) \) be the corresponding basic function field with... | Proof. Let \( \alpha \in \sum \) . We show by induction that the place \( \left( {{x}_{0} = \alpha }\right) \) splits completely in \( {F}_{n}/{F}_{0} \) for all \( n \geq 0 \) . This is trivial for \( n = 0 \), and we assume now that the assertion holds for some \( n \) . We have to show that every place \( Q \in {\ma... | Yes |
Corollary 7.2.21. Let \( \mathcal{F} = \left( {{F}_{0},{F}_{1},{F}_{2},\ldots }\right) \) be a tower over \( {\mathbb{F}}_{q} \) which is recursively defined by the equation \( f\left( Y\right) = h\left( X\right) \) . Let \( m \mathrel{\text{:=}} \deg f\left( Y\right) \) . Assume that \( \sum \subseteq {\mathbb{F}}_{q}... | Proof. Consider the basic function field \( F = {\mathbb{F}}_{q}\left( {x, y}\right) \) with defining equation \( f\left( y\right) = h\left( x\right) \) . Let \( P = \left( {x = \alpha }\right) \) with \( \alpha \in \sum \) be the place of \( {\mathbb{F}}_{q}\left( x\right) \) which is the zero of \( x - \alpha \), and... | Yes |
We return to the tower \( \mathcal{F} \) in Example 7.2.16; i.e., \( \mathcal{F} \) is recursively given by the equation \( f\left( Y\right) = h\left( X\right) \) with \( f\left( Y\right) = {Y}^{2} \) and \( h\left( X\right) = \left( {{X}^{2} + 1}\right) /{2X} \) over a field \( {\mathbb{F}}_{q} \) of odd characteristi... | The field \( {\mathbb{F}}_{9} \) can be represented as \( {\mathbb{F}}_{9} = {\mathbb{F}}_{3}\left( \delta \right) \) with \( {\delta }^{2} = - 1 \), so we have \( {\mathbb{F}}_{9} = \{ 0, \pm 1, \pm \delta , \pm \left( {\delta + 1}\right) , \pm \left( {\delta - 1}\right) \} \). We claim that the set \( \sum \mathrel{\... | Yes |
Proposition 7.2.23. Let \( \mathcal{F} = \left( {{F}_{0},{F}_{1},{F}_{2},\ldots }\right) \) be a recursive tower over \( {\mathbb{F}}_{q} \) , defined by the equation \( f\left( Y\right) = h\left( X\right) \), and let \( {\mathcal{F}}^{\prime } = \mathcal{F}L = \left( {{F}_{0}^{\prime },{F}_{1}^{\prime },{F}_{2}^{\prim... | Proof. By definition, the field \( {F}_{0}^{\prime } \) is the rational function field \( {F}_{0}^{\prime } = L\left( {x}_{0}\right) \) over \( L \) . Let \( P \in \operatorname{Ram}\left( {{\mathcal{F}}^{\prime }/{F}_{0}^{\prime }}\right) \) . There is some \( n \geq 0 \) and some place \( Q \) of \( {F}_{n}^{\prime }... | Yes |
We claim that \( \Lambda \) satisfies Condition (2) of Proposition 7.2.23. So we have to show that for all \( \beta \in \Lambda \), all solutions \( \alpha \in {\widetilde{\mathbb{F}}}_{q} \cup \{ \infty \} \) of the equation \( \left( {{\alpha }^{2} + 1}\right) /{2\alpha } = {\beta }^{2} \) are in \( \Lambda \) . | This is easily checked as follows:\n\n\[ \text{if}\beta = \infty \;\text{then}\alpha = 0\text{or}\alpha = \infty \text{;}\]\n\n\[ \text{if}\beta = 0\;\text{then}\alpha = \pm \delta \text{;}\]\n\n\[ \text{if}\beta = \pm 1\text{then}\alpha = 1\text{;}\]\n\n\[ \text{if}\beta = \pm \delta \text{then}\alpha = - 1\text{.} \] | Yes |
Theorem 7.3.1. Let \( m \geq 2 \) be an integer with \( q \equiv 1{\;\operatorname{mod}\;m} \) . Assume that the polynomial \( h\left( X\right) \in {\mathbb{F}}_{q}\left\lbrack X\right\rbrack \) has the following properties:\n\n(1) \( \deg h\left( X\right) = m \), and the leading coefficient of \( h\left( X\right) \) i... | Proof. We consider the sequence \( \mathcal{F} = \left( {{F}_{0},{F}_{1},{F}_{2},\ldots }\right) \), where \( {F}_{0} = {\mathbb{F}}_{q}\left( {x}_{0}\right) \) is a rational function field and for all \( n \geq 0 \) ,\n\n\[ \n{F}_{n + 1} = {F}_{n}\left( {x}_{n + 1}\right) \text{ with }{x}_{n + 1}^{m} = h\left( {x}_{n}... | Yes |
Proposition 7.3.2. Let \( q = {\ell }^{2} \) be a square, \( \ell > 2 \) . Then the equation\n\n\[ \n{Y}^{\ell - 1} = 1 - {\left( X + 1\right) }^{\ell - 1}\n\]\n\ndefines an asymptotically good tower \( \mathcal{F} \) over \( {\mathbb{F}}_{q} \) with limit\n\n\[ \n\lambda \left( \mathcal{F}\right) \geq 2/\left( {\ell -... | Proof. We set \( h\left( X\right) = 1 - {\left( X + 1\right) }^{\ell - 1} \) and \( \Lambda = {\mathbb{F}}_{\ell } \) . We need to check that the assumptions of Theorem 7.3.1 are satisfied:\n\n(1) The leading coefficient of \( h\left( X\right) \) is -1, which is a square in \( {\mathbb{F}}_{q} \) since \( q = {\ell }^{... | Yes |
Proposition 7.3.3. Let \( q = {\ell }^{e} \) with \( e \geq 2 \) . Then the equation\n\n\[ \n{Y}^{m} = 1 - {\left( X + 1\right) }^{m} \n\]\n\nwith \( m \mathrel{\text{:=}} \left( {q - 1}\right) /\left( {\ell - 1}\right) \) defines an asymptotically good tower \( \mathcal{F} \) over \( {\mathbb{F}}_{q} \) with limit\n\n... | Proof. In this case we have \( h\left( X\right) = 1 - {\left( X + 1\right) }^{m} \) and we set \( \Lambda \mathrel{\text{:=}} {\mathbb{F}}_{q} \) . Observe that the map \( \gamma \mapsto {\gamma }^{m} \) is the norm map from \( {\mathbb{F}}_{q} \) to \( {\mathbb{F}}_{\ell } \) and hence is surjective. Moreover, every e... | Yes |
Lemma 7.4.3. Equation (7.14) defines a recursive tower \( \mathcal{G} = \left( {{G}_{0},{G}_{1},{G}_{2},\ldots }\right) \) over \( {\mathbb{F}}_{q} \) . All extensions \( {G}_{i + 1}/{G}_{i} \) are Galois of degree \( \left\lbrack {{G}_{i + 1} : {G}_{i}}\right\rbrack = \ell \), and the place \( \left( {{x}_{0} = \infty... | Proof. It is clear that the equation \( {Y}^{\ell } - Y = {X}^{\ell }/\left( {1 - {X}^{\ell - 1}}\right) \) is separable and hence all extensions \( {G}_{i + 1}/{G}_{i} \) are separable of degree \( \left\lbrack {{G}_{i + 1} : {G}_{i}}\right\rbrack \leq \ell \) . Let \( {P}_{0} \mathrel{\text{:=}} \left( {{x}_{0} = \in... | Yes |
Lemma 7.4.4. Let \( \mathcal{G} \) be the tower over \( {\mathbb{F}}_{q} \) (with \( q = {\ell }^{2} \) ), which is recursively defined by Equation (7.14). Then the splitting locus of \( \mathcal{G}/{G}_{0} \) satisfies\n\n\[ \operatorname{Split}\left( {\mathcal{G}/{G}_{0}}\right) \supseteq \left\{ {\left( {{x}_{0} = \... | Proof. We want to show that the set \( \sum \mathrel{\text{:=}} {\mathbb{F}}_{q} \smallsetminus {\mathbb{F}}_{\ell } \) satisfies the condition of Corollary 7.2.21. So let \( \alpha \in \sum \) ; then\n\n\[ \frac{{\alpha }^{\ell }}{1 - {\alpha }^{\ell - 1}} \in {\mathbb{F}}_{q}\;\text{ since }{\alpha }^{\ell - 1} \neq ... | Yes |
Lemma 7.4.5. The tower \( \mathcal{G} \) over \( {\mathbb{F}}_{q} \) (with \( q = {\ell }^{2} \) ) which is recursively defined by Equation (7.14) has a finite ramification locus. More precisely one has \[ \operatorname{Ram}\left( {\mathcal{G}/{G}_{0}}\right) \subseteq \left\{ {\left( {{x}_{0} = \beta }\right) \mid \be... | It is easy to show that the tower \( \mathcal{G} \) is a wild tower. As we have seen in Lemma 7.4.3, in each step \( {G}_{i + 1}/{G}_{i} \) there are places which are totally (and hence wildly) ramified. So we do not have the estimate \( d\left( {Q \mid P}\right) \leq e\left( {Q \mid P}\right) \) for all \( P \in \oper... | No |
Proposition 7.4.13. Let \( E/F \) be a finite extension of function fields and let \( M, N \) be intermediate fields of \( E \supseteq F \) such that \( E = {MN} \) is the compositum of \( M \) and \( N \) . Assume that both extensions \( M/F \) and \( N/F \) are weakly ramified. Then \( E/F \) is weakly ramified. | Proof. The special case \( \left\lbrack {M : F}\right\rbrack = \left\lbrack {N : F}\right\rbrack = p \) has been considered in Proposition 3.9.4. The idea of proof here is to reduce the general case to this special case. There is a sequence of intermediate fields\n\n\[ F = {M}_{0} \subseteq {M}_{1} \subseteq \ldots \su... | Yes |
Lemma 7.4.14. Let \( K\left( x\right) \) be the rational function field over a field \( K \supseteq {\mathbb{F}}_{\ell } \) . Consider the subfields \( K\left( u\right) \subseteq K\left( t\right) \subseteq K\left( x\right) \) with\n\n\[ t \mathrel{\text{:=}} {x}^{\ell } - x\text{ and }u \mathrel{\text{:=}} {\left( {x}^... | Proof. (a) We consider the two subgroups \( {U}_{0},{U}_{1} \) of the automorphism group of \( K\left( x\right) /K \) which are defined by\n\n\[ {U}_{0} \mathrel{\text{:=}} \left\{ {\sigma : x \mapsto {ax} + b \mid a \in {\mathbb{F}}_{\ell }^{ \times }, b \in {\mathbb{F}}_{\ell }}\right\} ,\]\n\n\[ {U}_{1} \mathrel{\te... | No |
Theorem 7.4.15. Let \( q = {\ell }^{2} \) be a square. With the above notation, the following hold:\n\n(a) \( {\mathbb{F}}_{q} \) is the full constant field of \( {G}_{i}^{ * } \) for all \( i \geq 0 \) . Therefore the sequence\n\n\[ \n{\mathcal{G}}^{ * } \mathrel{\text{:=}} \left( {{G}_{0}^{ * },{G}_{1}^{ * },{G}_{2}^... | Proof. We consider the field extensions\n\n\[ \n{\mathbb{F}}_{q}\left( {u}_{0}\right) \subseteq {\mathbb{F}}_{q}\left( {t}_{0}\right) \subseteq {\mathbb{F}}_{q}\left( {x}_{0}\right) = {G}_{0} \subseteq {G}_{1} \subseteq {G}_{2} \subseteq \ldots\n\]\n\n(7.28)\n\nThe first step \( {\mathbb{F}}_{q}\left( {t}_{0}\right) /{... | Yes |
Corollary 7.4.16. With the notations of Theorem 7.4.15, we set\n\n\[ \n{n}_{i} \mathrel{\text{:=}} \left\lbrack {{G}_{i}^{ * } : {\mathbb{F}}_{q}\left( {u}_{0}\right) }\right\rbrack = \left( {\ell - 1}\right) \cdot {m}_{i} \]\n\n(7.30)\n\nfor every \( i \geq 0 \), so \( {m}_{i} = \left\lbrack {{G}_{i}^{ * } : {\mathbb{... | Proof. The places \( \left( {{t}_{0} = 0}\right) \) and \( \left( {{t}_{0} = \infty }\right) \) are the only places of \( {\mathbb{F}}_{q}\left( {t}_{0}\right) \) which ramify in the extension \( {G}_{i}^{ * }/{\mathbb{F}}_{q}\left( {t}_{0}\right) \) . Since they are weakly ramified (Theorem 7.4.15(c),(d)), the differe... | Yes |
Lemma 7.4.19. Assume that \( {\mathbb{F}}_{\ell } \subseteq K \) . With the above notation we have:\n\n(a) The extensions \( K\left( x\right) /K\left( u\right), K\left( y\right) /K\left( u\right), K\left( y\right) /K\left( v\right) \) and \( K\left( z\right) /K\left( v\right) \) are Galois of degree \( \ell \left( {\el... | Proof. (a) By Lemma 7.4.14(a), the extension \( K\left( y\right) /K\left( u\right) \) is Galois of degree \( \left\lbrack {K\left( y\right) : K\left( u\right) }\right\rbrack = \ell \left( {\ell - 1}\right) \), since \( u = {\left( {y}^{\ell } - y\right) }^{\ell - 1} + 1 \) . Now we observe that the equation \( u = - {x... | Yes |
Lemma 7.4.20. We maintain the notation of Equations (7.36), (7.37) and Figures 7.3,7.4, and also assume that \( {\mathbb{F}}_{{\ell }^{2}} \subseteq K \) . Let \( R \) be a place of \( H \) which is ramified in the extension \( H/K\left( u\right) \) . Then the restriction \( {P}^{ * } = R \cap K\left( u\right) \) of \(... | Proof. Since \( R \\mid {P}^{ * } \) is ramified, at least one of the places \( P \\mid {P}^{ * } \) or \( {R}^{ * } \\mid {P}^{ * } \) is ramified, see Figure 7.4. We distinguish several cases:\n\n(i) Assume that \( P \\mid {P}^{ * } \) is ramified. Since \( K\\left( x\\right) = K\\left( {1/x}\\right) \) and\n\n\[ {\\... | Yes |
Proposition 7.4.22. Let \( K \) be a field with \( {\mathbb{F}}_{\ell } \subseteq K \), and consider the sequence \( \mathcal{H} = \left( {{H}_{0},{H}_{1},{H}_{2},\ldots }\right) \) where \( {H}_{0} = K\left( {y}_{0}\right) \) is a rational function field, and \( {H}_{i + 1} = {H}_{i}\left( {y}_{i + 1}\right) \) with\n... | Proof. The field \( {H}_{2} = K\left( {{y}_{0},{y}_{1},{y}_{2}}\right) \) is isomorphic to the function field \( H = \) \( K\left( {x, y, z}\right) \) that we studied in Lemmas 7.4.19 and 7.4.20. Therefore we know already that \( {H}_{1}/{H}_{0} \) is Galois of degree \( \ell \left( {\ell - 1}\right) \), the extension ... | Yes |
Lemma 7.4.24. Assume that \( {\mathbb{F}}_{{\ell }^{2}} \subseteq K \) . Then the ramification locus of \( \mathcal{H} \) over \( {H}_{0} \) satisfies\n\n\[ \operatorname{Ram}\left( {\mathcal{H}/{H}_{0}}\right) \subseteq \left\{ {\left( {{y}_{0} = \beta }\right) \mid \beta \in {\mathbb{F}}_{{\ell }^{2}}\cup \{ \infty \... | Proof of Lemma 7.4.24. We want to apply Proposition 7.2.23. We set\n\n\[ {\Lambda }_{0} \mathrel{\text{:=}} \left\{ {{y}_{0}\left( P\right) \mid P \in {\mathbb{P}}_{{H}_{0}}\text{ is ramified in }{H}_{1}/{H}_{0}}\right\} . \]\n\nThen \( {\Lambda }_{0} = {\mathbb{F}}_{{\ell }^{2}}^{ \times } \cup \{ \infty \} \) by Lemm... | Yes |
Corollary 7.4.25. The genus \( \gamma \left( {\mathcal{H}/{H}_{0}}\right) \) of the tower \( \mathcal{H} \) is finite; it is bounded \( {by} \)\n\n\[ \gamma \left( {\mathcal{H}/{H}_{0}}\right) \leq \frac{{\ell }^{2} + 2\ell }{2}. \] | Proof. Recall that both extensions \( {H}_{2}/K\left( {{y}_{0},{y}_{1}}\right) \) and \( {H}_{2}/K\left( {{y}_{1},{y}_{2}}\right) \) are weakly ramified Galois extensions of degree \( \ell \) by Corollary 7.4.21. For all \( n \geq 2 \) we consider the field \( {H}_{n} \) as the compositum of the fields \( {H}_{n - 1} \... | Yes |
Proposition 8.1.1.\n\n\[ \n{C}_{\Omega }\left( {D, G}\right) = \left\{ {\left( {{\operatorname{res}}_{{P}_{1}}\left( \omega \right) ,\ldots ,{\operatorname{res}}_{{P}_{n}}\left( \omega \right) }\right) \mid \omega \in {\Omega }_{F}\left( {G - D}\right) }\right\} .\n\] | It is this representation that is most commonly used in the literature to define the code \( {C}_{\Omega }\left( {D, G}\right) \) .\n\nH. Stichtenoth, Algebraic Function Fields and Codes,\n\n289\n\nGraduate Texts in Mathematics 254,\n\n(C) Springer-Verlag Berlin Heidelberg 2009\n\nBy Proposition 2.2.10 the code \( {C}_... | No |
Proposition 8.1.2. Let \( t \) be an element of \( F \) such that \( {v}_{{P}_{i}}\left( t\right) = 1 \) for \( i = 1,\ldots, n \) . Then the following hold:\n\n(a) The differential \( \eta \mathrel{\text{:=}} {dt}/t \) satisfies \( {v}_{{P}_{i}}\left( \eta \right) = - 1 \) and \( {\operatorname{res}}_{{P}_{i}}\left( \... | Proof. (a) Since \( t \) is a prime element of \( P \mathrel{\text{:=}} {P}_{i} \), the \( P \) -adic power series of \( \eta = {dt}/t \) with respect to \( t \) is\n\n\[ \eta = \frac{1}{t}{dt} \]\n\nHence \( {v}_{P}\left( \eta \right) = - 1 \) and \( {\operatorname{res}}_{P}\left( \eta \right) = 1 \) .\n\n(b) Follows ... | Yes |
Corollary 8.1.3. Suppose that \( t \in F \) is a prime element for all places \( {P}_{1},\ldots ,{P}_{n} \)\n\n(a) If \( {2G} - D \leq \left( {{dt}/t}\right) \) then the code \( {C}_{\mathcal{L}}\left( {D, G}\right) \) is self-orthogonal; i.e.,\n\n\[ {C}_{\mathcal{L}}\left( {D, G}\right) \subseteq {C}_{\mathcal{L}}{\le... | Proof. This is an immediate consequence of Corollary 2.2.11. | No |
Proposition 8.2.3. (a) \( {\operatorname{Aut}}_{D, G}\left( {F/{\mathbb{F}}_{q}}\right) \) acts on the code \( {C}_{\mathcal{L}}\left( {D, G}\right) \) by\n\n\[ \sigma \left( \left( {x\left( {P}_{1}\right) ,\ldots, x\left( {P}_{n}\right) }\right) \right) \mathrel{\text{:=}} \left( {x\left( {\sigma \left( {P}_{1}\right)... | Proof. (a) We begin with the following assertion: given a place \( P \) of degree one and an element \( y \in F \) with \( {v}_{P}\left( y\right) \geq 0 \), we have\n\n\[ \sigma \left( y\right) \left( {\sigma \left( P\right) }\right) = y\left( P\right) \]\n\n(8.3)\n\nIn fact, setting \( a \mathrel{\text{:=}} y\left( P\... | Yes |
Example 8.2.4. As an example we consider a BCH code \( C \) of length \( n \) over \( {\mathbb{F}}_{q} \) . As shown in Section 2.3, \( C \) can be realized as a subfield subcode of a rational AG code as follows: let \( n \mid \left( {{q}^{m} - 1}\right) \) and let \( \beta \in {\mathbb{F}}_{{q}^{m}} \) be a primitive ... | \[ C = {\left. {C}_{\mathcal{L}}\left( {D}_{\beta }, r{P}_{0} + s{P}_{\infty }\right) \right| }_{{\mathbb{F}}_{q}} \] with \( r, s \in \mathbb{Z} \) (see Proposition 2.3.9). The automorphism \( \sigma \in \operatorname{Aut}\left( {F/{\mathbb{F}}_{{q}^{m}}}\right) \) given by \( \sigma \left( z\right) = {\beta }^{-1}z \... | Yes |
Proposition 8.3.2. The dual code of \( {C}_{r} \) is\n\n\[ \n{C}_{r}^{ \bot } = {C}_{{q}^{3} + {q}^{2} - q - 2 - r}.\n\]\n\nHence \( {C}_{r} \) is self-orthogonal if \( {2r} \leq {q}^{3} + {q}^{2} - q - 2 \), and \( {C}_{r} \) is self-dual for \( r = \left( {{q}^{3} + {q}^{2} - q - 2}\right) /2 \) . | Proof. Consider the element\n\n\[ \nt \mathrel{\text{:=}} \mathop{\prod }\limits_{{\alpha \in {\mathbb{F}}_{{q}^{2}}}}\left( {x - \alpha }\right) = {x}^{{q}^{2}} - x.\n\]\n\n\( t \) is a prime element for all places \( {P}_{\alpha ,\beta } \leq D \), and its principal divisor is \( \left( t\right) = D - {q}^{3}{Q}_{\in... | Yes |
Proposition 8.3.3. Suppose that \( 0 \leq r \leq {q}^{3} + {q}^{2} - q - 2 \) . Then the following hold:\n\n(a) The dimension of \( {C}_{r} \) is given by\n\n\[ \dim {C}_{r} = \left\{ \begin{matrix} \left| {I\left( r\right) }\right| & \text{ for }0 \leq r < {q}^{3}, \\ {q}^{3} - \left| {I\left( s\right) }\right| & \tex... | Proof. (a) For \( 0 \leq r < {q}^{3} \) Corollary 2.2.3 gives\n\n\[ \dim {C}_{r} = \dim \mathcal{L}\left( {r{Q}_{\infty }}\right) = \left| {I\left( r\right) }\right| . \]\n\nFor \( {q}^{3} \leq r \leq {q}^{3} + {q}^{2} - q - 2 \) we set \( s \mathrel{\text{:=}} {q}^{3} + {q}^{2} - q - 2 - r \) . Then \( 0 \leq s \leq \... | Yes |
Corollary 8.3.4. Suppose that \( 0 \leq r < {q}^{3} \) . Let \( 0 = {s}_{1} < {s}_{2} < \ldots < {s}_{k} \leq r \) be all pole numbers \( \leq r \) of \( {Q}_{\infty } \) . Then the \( k \times {q}^{3} \) matrix \( {M}_{r} \) whose rows are \( {u}_{{s}_{1}},\ldots ,{u}_{{s}_{k}} \), is a generator matrix of \( {C}_{r} ... | Proof. Corollary 2.2.3. | No |
Proposition 8.4.2. There is a continuous function \( {\alpha }_{q} : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) such that\n\n\[ \n{U}_{q} = \left\{ {\left( {\delta, R}\right) \mid 0 \leq \delta \leq 1\text{ and }0 \leq R \leq {\alpha }_{q}\left( \delta \right) }\right\} .\n\]\n\nMore... | The proof of this proposition requires only elementary techniques of coding theory; we refer to [29]. | No |
Proposition 8.4.4 (Gilbert-Varshamov Bound). For \( 0 \leq \delta \leq 1 - {q}^{-1} \) , \[ {\alpha }_{q}\left( \delta \right) \geq 1 - {H}_{q}\left( \delta \right) \] | The Gilbert-Varshamov bound is the best lower bound for \( {\alpha }_{q}\left( \delta \right) \) which is known from elementary coding theory. However, its proof is not constructive (i.e., it does not provide a simple algebraic algorithm for the construction of good long codes). | No |
Lemma 8.4.5. Suppose that \( {P}_{1},\ldots ,{P}_{n} \) are distinct places of \( F/{\mathbb{F}}_{q} \) of degree one. Then there exists, for each \( r \geq 0 \), a divisor \( G \) such that \( \deg G = r \) and \( {P}_{i} \notin \operatorname{supp}G \) (for \( i = 1,\ldots, n \) ). | Proof. The lemma is trivial if there is another place \( Q \) of degree one, different from \( {P}_{1},\ldots ,{P}_{n} \) . In this case we set \( G \mathrel{\text{:=}} {rQ} \) . If \( {P}_{1},\ldots ,{P}_{n} \) are all the places of \( F/{\mathbb{F}}_{q} \) of degree one, we choose a divisor \( G \sim r{P}_{1} \) (i.e... | Yes |
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