Class numbers of some abelian extensions of rational function fields

Abstract

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

Determinant formulas for class numbers in function fields

“…Recently the authors gave determinant formulas for the real and relative class number of any subfield of a cyclotomic field with prime power conductor [2]. In this paper, by extending Kučera's idea [8,Lemma 2] to the function field case, we obtain several determinant formulas involving the real class number and the relative class number of any subfield of cyclotomic function fields with arbitrary conductor.…”