Class numbers of some abelian extensions of rational function fields

Abstract

Let P be a monic irreducible polynomial. In this paper we generalize the determinant formula for h(K+pn) of Bae and Kang and the formula for h-(Kpn) of Jung and Ahn to any subfields K of the cyclotomic function field Kpn. By using these formulas, we calculate the class numbers h-(K), h(k+) of all subfields K of Kp when q and deg(P) are small.