Determining irreducibility and ramification groups for an additive extension of the rational function field.
type: Item
instance of: scholarly article
title: Determining irreducibility and ramification groups for an additive extension of the rational function field.
zbMATH Open document ID: 1036.11060
DOI: 10.1016/S0022-314X(02)00012-4
author: Vinay Deolalikar
published in: Journal of Number Theory
publication date: 2003-04-02T00:00:00Z
review text: Let \(K\) be a subfield of \(\overline{\mathbb F}_p\), not necessarily proper. If \(V\) is a finite subgroup of the additive group of \(K\), let \(a_V(T)\) denote the additive polynomial \(\prod_{v\in V} (T-v)\). The author considers the extension of \(K(x)\) obtained by adjoining a root of \(a_V(T)-f(x)\), where \(f(x)\in K[x]\). First, the following criterion for the irreducibility of \(a_V(T)-f(x)\) is given. A term in \(f(x)\) is called a coprime term if its degree is coprime to \(p\), and a coprime term of degree \(d\) is called totally coprime if \(f(x)\) has no terms of degree \(dp^i\) for \(i>0\). The coprime degree of \(f(x)\) is the degree of the largest totally coprime term if \(f(x)\) has any totally coprime terms, and is zero otherwise. The author proves that if the coprime degree of \(f(x)\) is positive, then the polynomial \(a_V(T)-f(x)\) is irreducible over \(K(x)\) for every finite subgroup \(V\) of the additive group of \(K\). In many cases, this irreducibility criterion is simpler to apply than the Artin-Schreier Theorem. If the coprime degree of \(f\) is greater than \({\deg}(f)/p\), then the author determines the sequence of ramification groups at the place at infinity and gives the genus of the above extension of \(K(x)\). This theory is applied when \(K={\mathbb F}_{q^n}, a_V(T)=\sum_{0\leq j\leq n-1}T^{q^j}\), and \(f(x)\) is an \((n,q)\)-elementary symmetric polynomial, a type of polynomial that also has been considered by \textit{A. Garcia} and \textit{H. Stichtenoth} [Finite Fields Appl. 5, 424--435 (1999; Zbl 0965.11027)]. The author computes the number of rational places of the extension in this situation, and it is shown that the extension has relatively many rational places when \(f(x)=\sum x^{q^j+q^k}\), where \(j<k\) and \(0\leq j,k\leq n-1\). An appendix contains more details on \((n,q)\)-symmetric polynomials.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11G20
zbMATH DE Number: 1891442
zbMATH Keywords: additive polynomial
zbMATH Keywords: irreducibility
zbMATH Keywords: ramification groups
zbMATH Keywords: function fields with many rational places
reviewed by: Robert F. Lax
MaRDI profile type: Publication
cites work: EXTENSIONS OF ALGEBRAIC FUNCTION FIELDS WITH COMPLETE SPLITTING OF ALL RATIONAL PLACES
cites work: Q5690779
cites work: A class of polynomials over finite fields
cites work: Q4368843
cites work: Q3324076
cites work: Q3856855
cites work: Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I: Eine Abschätzung der Ordnung der Automorphismengruppe
cites work: Algebraic function fields and codes
cites work: Q3997667
description: scientific article; zbMATH DE number 1891442