Download Algebraic Function Fields and Codes by Henning Stichtenoth PDF
By Henning Stichtenoth
The idea of algebraic functionality fields has its origins in quantity thought, complicated research (compact Riemann surfaces), and algebraic geometry. given that approximately 1980, functionality fields have chanced on wonderful purposes in different branches of arithmetic similar to coding thought, cryptography, sphere packings and others. the most goal of this ebook is to supply a simply algebraic, self-contained and in-depth exposition of the idea of functionality fields.
This new version, released within the sequence Graduate Texts in arithmetic, has been significantly extended. in addition, the current variation includes quite a few routines. a few of them are relatively effortless and support the reader to appreciate the fundamental fabric. different workouts are extra complex and canopy extra fabric that can now not be incorporated within the text.
This quantity is especially addressed to graduate scholars in arithmetic and theoretical computing device technology, cryptography, coding concept and electric engineering.
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Extra resources for Algebraic Function Fields and Codes
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. Proof. Suppose that deg B = 2g − 2 and (B) ≥ g. Choose a canonical divisor W . Then g ≤ (B) = deg B + 1 − g + (W − B) = g − 1 + (W − B) , therefore (W − B) ≥ 1. 12 that W ∼ B. Next we come to a characterization of the rational function ﬁeld. 3. , F = K(x) for some x which is transcendental over the ﬁeld K.
Then CL (D, G)⊥ = CΩ (D, G) = CL (D, H) with H := D − G + (η) . Proof. 8. Observe that supp (D − G + (η)) ∩ supp D = ∅ since vPi (η) = −1 for i = 1, . . , n. Hence the code CL (D, D − G + (η)) is deﬁned. 14 there is an isomorphism μ : L (D − G + (η)) → ΩF (G − D) given by μ(x) := xη. For x ∈ L (D − G + (η)) we have (xη)Pi (1) = ηPi (x) = x(Pi ) · ηPi (1) = x(Pi ), cf. 10). This implies CΩ (D, G) = CL (D, D − G + (η)). 11. Suppose there is a Weil diﬀerential η such that 2G − D ≤ (η) and ηPi (1) = 1 for i = 1, .
7, this implies i(A) = (W − A). Summing up the results of this section we obtain the Riemann-Roch Theorem; it is by far the most important theorem in the theory of algebraic function ﬁelds. 15 (Riemann-Roch Theorem). Let W be a canonical divisor of F/K. Then for each divisor A ∈ Div(F ), (A) = deg A + 1 − g + (W − A) . 6 Some Consequences of the Riemann-Roch Theorem 31 Proof. 14 and the deﬁnition of i(A). 16. For a canonical divisor W we have deg W = 2g − 2 and (W ) = g . Proof. 7 give 1 = (0) = deg 0 + 1 − g + (W − 0) .