 
Summary: Class number in non Galois quartic and non
abelian Galois octic function fields over finite
fields
Yves Aubry
G. R. I. M.
Universit´e du Sud ToulonVar
83 957 La Garde Cedex
France
yaubry@univtln.fr
Abstract
We consider a totally imaginary extension of a real extension of a rational function
field over a finite field of odd characteristic. We prove that the relative ideal class
number one problem for such non Galois quartic fields is equivalent to the one for
non abelian Galois octic imaginary functions fields. Then, we develop some results
on characters which give a method to evaluate the ideal class number of such quartic
function fields.
1 Introduction
Let L be a totally imaginary extension of a function field K which is itself a real extension
of a rational function field k = Fq(x). This means that the infinite place of k totally
splits in the extension K/k and that this places have only one place above each of them in
