Summary: Imaginary bicyclic biquadratic function fields in
Yves Aubry and Dominique Le Brigand
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 Xianke. Precisely, we study the ideal
class number one problem for imaginary quartic Galois extensions of
k = Fq(x) of Galois group Z/2Z × Z/2Z in even characteristic.
Let L/k be such an extension and let K1, K2 and K3 be the distinct
subfields extensions of L/k. In even characteristic, the fields Ki are
Artin-Schreier extensions of k and L is the compositum of any two of
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 extensions L/k of Galois group Z/2Z× Z/2Z
with ideal class number one.
Key words - Ideal class number, function fields, Artin-Schreier ex-
tensions, zeta functions.