A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound

Summary For an algebraic function fieldF having a finite constant field, letg(F) (resp.N(F)) denote the genus ofF (resp. the number of places ofF of degree one). We construct a tower of function fields over such that the ratioN(F _i )/g(F _i ) tends to the Drinfeld-Vladut boundq–1.Oblatum 5-XII-1994 @ 2-II-95This paper was written when the second author was visiting the Instituto de Matemática Pura e Aplicada, Rio de Janeiro (July–Sept. 1994). This visit was supported by BMFT and CNPq.This article was processed by the author using the L^AT _EX style filepljour1m from Springer-Verlag.     

Artin-Schreier extensions and Galois module structure

Let k be the power series field over a finite field of characteristic p>0. Let L be a cyclic extension over k of degree p. We give a necessary and sufficient condition for the integer ring O_L to be free over the associated order. Moreover, when O_L is free, we construct a free basis explicitly.     

Nonassociative cyclic extensions of fields and central simple algebras

We define nonassociative cyclic extensions of degree m of both fields and central simple algebras over fields. If a suitable field contains a primitive mth (resp., qth) root of unity, we show that suitable nonassociative generalized cyclic division algebras yield nonassociative cyclic extensions of degree m (resp., qs). Some of Amitsur's classical results on non-commutative associative cyclic extensions of both fields and central simple algebras are obtained as special cases.     

The discriminant problem for Artin-Schreier extensions: Explicit results

Given a global function field K of characteristic p, for all effective divisors $\mathfrak{D}$ in the divisor group of K we count the number of cyclic extensions $F\supset K$ of degree p where the relative discriminant ${\mathrm{\text{Disc}}}_K(F)=(p-1)\mathfrak{D}$.     

On towers of function fields of Artin-Schreier type

In this article we derive strong conditions on the defining equations of asymptotically good Artin-Schreier towers. We will show that at most three kinds of defining equations can give rise to a recursively defined good tower, if we restrict ourselves to prime degrees. ¹A. Garcia and H. Stichtenoth did part of thiswork during their stay at Sabanci University, Istanbul, Turkey (Sept. 2002). ²A. Garcia was partially supported by PRONEX # 662408/1996-3 (CNPq-Brazil).     

Arithmetic in generalized Artin-Schreier extensions of k(x)

If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions K_n over k(x) such that [K : k(x)] = pⁿ using the concept of Witt vectors. This is accomplished in the following way; if [β₁, β₂,…, β_n] is a Witt vector over k(x) = K₀, then the Witt equation $\text{y}{}^{\mathrm{p}}\text{?}\text{y}\text{= β}$ generates a tower of extensions through $\text{K}{}_{\mathrm{i}}\text{= K}{}_{\mathrm{i?1}}\text{(}\text{y}{}_{\mathrm{i}}\text{)}$ where $\text{y}\text{= [}\text{y}{}_1\text{,}\text{y}{}_2\text{,…,}\text{y}{}_{\mathrm{n}}\text{]}$. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K_n. This alternate generation has the form K_i = K_i?1(y_i); y_i^p ? y_i = B_i, where, as a divisor in K_i?1, B_i has the form . In this form q is prime to Πp_j^λ_j and each λ_j is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K_n over an algebraically closed field of constants.     

Some Artin-Schreier type function fields over finite fields with prescribed genus and number of rational places

We give existence and characterization results for some Artin-Schreier type function fields over finite fields with prescribed genus and number of rational places simultaneously.     

G-primitive extensions of differential fields

A classification is given of the G-primitive extensions of a partial differential field of characteristic zero with an algebraically closed field of constants and a connected algebraic group G.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 425–434, March, 1978.     

Retract rational fields and cyclic Galois extensions

In [23], this author began a study of so-called lifting and approximation problems for Galois extensions. One primary point was the connection between these problems and Noether’s problem. In [24], a similar sort of study was begun for central simple algebras, with a connection to the center of generic matrices. In [25], the notion of retract rational field extension was defined, and a connection with lifting questions was claimed, which was used to complete the results in [23] and [24] about Noether's problem and generic matrices. In this paper we, first of all, set up a language which can be used to discuss lifting problems for very general “linear structures”. Retract rational extensions are defined, and proofs of their basic properties are supplied, including their connection with lifting. We also determine when the function fields of algebraic tori are retract rational, and use this to further study Noether’s problem and cyclic 2-power Galois extensions. Finally, we use the connection with lifting to show that ifp is a prime, then the center of thep degree generic division algebra is retract rational over the ground field. The author is grateful for NSF support under grant #MCS79-04473.     

Valuation rings and simple transcendental field extensions

If a valuation ring V on a simple transcendental field extension K₀(X) is such that the residue field k of V is not algebraic over the residue field k₀ of V₀=V∩K₀, then for k₀ a perfect field it is shown that k is obtained from k₀ by a finite algebraic followed by a simple transcendental field extension.     

Genus fields of Kummer extensions of rational function fields

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.