GENUS FIELDS OF CYCLIC l-EXTENSIONS OF RATIONAL FUNCTION FIELDS

Abstract

We give a construction of genus fields for Kummer cyclic l-extensions of rational congruence function fields, l a prime number. First we find this genus field for a field contained in a cyclotomic function field using Leopoldt's construction by means of Dirichlet characters and the Hilbert class field defined by Rosen. The general case follows from this. This generalizes the result obtained by Peng for a cyclic extension of degree l.

Genus fields of congruence function fields

“…We have e P∞ = e ∞ = l n−d , where d = min{n, d ′ } and v l (deg D) = d ′ . Furthermore the inertia degree of P ∞ is f ∞ = l m , where F q l m = F q ( l d (−1) deg D γ) (see [4,Proposition 2.8]). Hence t 0 = l m and the field of constants of K ge is F q l m .…”

“…The genus field K ge was obtained for an abelian extension K of k. The method can be used to give K ge explicitly when K/k is a cyclic extension of prime degree l | q − 1 (Kummer) or l = p where p is the characteristic (Artin-Schreier) and also when K/k is a p-cyclic extension (Witt). Later on, the method was used in [4] to describe K ge explicitly when K/k is a cyclic extension of degree l n , where l is a prime number and l n | q − 1.…”

Function field genus theory for non-Kummer extensions

“…Using Leopoldt's technique we applied Dirichlet characters to the function field case and found a general description of K ge ( [1,4,5]). In these papers it was also provided an explicit description of K ge in the cases of a Kummer cyclic extension of prime degree l and of an abelian p-extension where p is the characteristic of k. In [2,6,8], the explicit description of K ge was given when K/k is a finite Kummer l-extension with l a prime number.…”

Genus fields of Kummer extensions of rational function fields

“…This paper can be considered as the end of the problem first studied by Peng [7] where he found the genus field of a cyclic Kummer extension of a rational global function field of degree l. The next step was the case of a Kummer cyclic extension of prime power degree. This problem was first studied in [2] under the condition that the field was contained in a cyclotomic function field and a strong restriction. In [8] the general Kummer cyclic extension of prime power degree was settled.…”