Imaginary Bicyclic Biquadratic Function Fields in Characteristic Two

Abstract

We are interested in the analogue of a result proved in the number field case by E. Brown and C. J. Parry and in the function field case in odd characteristic by Zhang Xian-Ke. Precisely, we study the ideal class number one problem for imaginary quartic Galois extensions ofk=Fq(x) of Galois group Z/2Z×Z/2Z in even characteristic. LetL/kbe such an extension and letK1,K2, andK3be the distinct subfields extensions ofL/k. In even characteristic, the fieldsKiare Artin–Schreier extensions ofkandLis the compositum of any two of them. Using the factorization of the zeta functions of this fields, we get a formula between their ideal class numbers which enables us to find all imaginary quartic Galois extensionsL/kof Galois group Z/2Z×Z/2Z with ideal class number one.