Algebraic Function Fields and Codes (Graduate Texts in by Henning Stichtenoth PDF

By Henning Stichtenoth

ISBN-10: 3540768777

ISBN-13: 9783540768777

This booklet hyperlinks topics: algebraic geometry and coding thought. It makes use of a unique technique according to the idea of algebraic functionality fields. insurance contains the Riemann-Rock theorem, zeta capabilities and Hasse-Weil's theorem in addition to Goppa' s algebraic-geometric codes and different conventional codes. it will likely be invaluable to researchers in algebraic geometry and coding concept and machine scientists and engineers in info transmission.

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Additional resources for Algebraic Function Fields and Codes (Graduate Texts in Mathematics, Volume 254)

Sample text

33) 32 1 Foundations of the Theory of Algebraic Function Fields Proof. Setting A = 0 resp. 33) we obtain (W0 ) = g0 and deg W0 = 2g0 − 2 (cf. 16). Let W be a canonical divisor of F/K. We choose a divisor A with deg A > max{2g − 2, 2g0 − 2}. 33). Therefore g = g0 . 33). This yields g = (2g − 2) + 1 − g + (W0 − W ) , hence (W0 − W ) = 1. Since deg(W0 − W ) = 0, this implies that W0 − W is principal (cf. 12), so W0 ∼ W . Another useful characterization of canonical divisors is the following. 2. A divisor B is canonical if and only if deg B = 2g − 2 and (B) ≥ g.

The crucial step of the proof is the following claim: Claim. Given g distinct places P1 , . . , Pg ∈ T and a divisor A ≥ 0 with (A) = 1 and deg A ≤ g − 1, there is an index j ∈ {1, . . , g} such that (A + Pj ) = 1 . Suppose the claim is false; then (A + Pj ) > 1, and there are elements zj ∈ L (A + Pj )\L (A) for j = 1, . . , g. Since vPj (zj ) = −vPj (A) − 1 and vPi (zj ) ≥ −vPi (A) for i = j , 36 1 Foundations of the Theory of Algebraic Function Fields the Strict Triangle Inequality implies that the g + 1 elements 1, z1 , .

Now we consider a function field F/K, a divisor A ∈ Div(F ) and a non-zero Weil differential ω ∈ ΩF . Let W := (ω). Show that the map s : L (W − A) × AF /(AF (A) + F ) → K given by s(x, α) := ω(xα) is well-defined, and it is a non-degenerate pairing. 15. Assume that the constant field K is algebraically closed. Show that for every integer d ≥ g, there exists a divisor A ∈ Div(F ) with deg(A) = d and (A) = deg(A) + 1 − g. 16. Let i(A) denote the index of specialty of the divisor A ∈ Div(F ). Show: (i) i(A) ≤ max{0, 2g − 1 − deg(A)}.

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Algebraic Function Fields and Codes (Graduate Texts in Mathematics, Volume 254) by Henning Stichtenoth


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