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

By Henning Stichtenoth

This e-book hyperlinks matters: algebraic geometry and coding thought. It makes use of a unique procedure according to the speculation of algebraic functionality fields. insurance comprises the Riemann-Rock theorem, zeta features and Hasse-Weil's theorem in addition to Goppa' s algebraic-geometric codes and different conventional codes. it is going to be precious to researchers in algebraic geometry and coding conception and machine scientists and engineers in details transmission.

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Iii) F/IR is not a rational function field. (iv) All places of F/IR have degree 2. 11. Assume that char(K) = 2. Let F = K(x, y) with y 2 = f (x) ∈ K[x] , deg f (x) = 2m + 1 ≥ 3 . Show: (i) K is the full constant field of F . (ii) There is exactly one place P ∈ IPF which is a pole of x, and this place is also the only pole of y. (iii) For every r ≥ 0, the elements 1, x, x2 , . . , xr , y, xy, . . , xs y with 0 ≤ s < r − m are in L (2rP ). (iv) The genus of F/K satisfies g ≤ m. Remark. 6. 12. Let K = IF3 be the field with 3 elements and K(x) the rational function field over K.

Let V, W be vector spaces over K. A non-degenerate pairing of V and W is a bilinear map s : V × W → K such that the following hold: For every v ∈ V with v = 0 there is some w ∈ W with s(v, w) = 0, and for every w ∈ W with w = 0 there is some v ∈ V with s(v, w) = 0. 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.

5 The Riemann-Roch Theorem In this section F/K denotes an algebraic function field of genus g. 1. For A ∈ Div(F ) the integer i(A) := (A) − deg A + g − 1 is called the index of specialty of A. 17 states that i(A) is a non-negative integer, and i(A) = 0 if deg A is sufficiently large. In the present section we will provide several interpretations for i(A) as the dimension of certain vector spaces. To this end we introduce the notion of an adele. 2. An adele of F/K is a mapping α: IPF P −→ −→ F, αP , such that αP ∈ OP for almost all P ∈ IPF .

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