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

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, $p \geq 3$ prime, and degree $2d = [K:\mathbb{Q}] \leq 10$, in terms of sums of 2d-power residues. In particular, tables are given for $p \leq 10000$.