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 greater-than-or-equal-to 3 prime, and degree 2d = [K: Q] less-than-or-equal-to 10, in terms of sums of 2d-power residues. In particular, tables are given for p less-than-or-equal-to 1 0000 .