The relative class numbers of imaginary cyclic fields of degrees 4,6,8, and 10

Abstract

We express the relative class number of an imaginary abelian number field K of prime power conductor as a sort of Maillet determinant. Thereby we obtain explicit relative class number formulas for fields K of conductor p, p ≥ 3 p \geq 3 prime, and degree 2 d = [ K : Q ] ≤ 10 2d = [K:\mathbb {Q}] \leq 10 , in terms of sums of 2d-power residues. In particular, tables are given for p ≤ 10000 p \leq 10000 .