The relative class numbers of certain imaginary abelian number fields and determinants

AbstractLet p be an odd prime. For a rational integer a, denote by R′(a) a rational integer that satisfies −(p − 1)2 ≤ R′(a) ≤ (p − 1)2 and R′(a) ≡ a (mod p). We define a determinant D′p by D′p = |R′(ab′)|1 ≤ a, b ≤ (p − 1)2, where bb′ ≡ 1 (mod p). It is shown that D′p is related to the relative class numbers of imaginary subfields of the 8pth cyclotomic number field, and, if p ≡ ±3 (mod 8), D′p ≠ 0. The other determinants are also studied