Congruences modulo 8 for class numbers of general quadratic fields Q(√m) and Q(√−m)

AbstractCongruences modulo 8 for class numbers h and h∗ of Q(√m) and Q(√−m) are obtained, 3 < m ∈ Z squarefree. Let ε = t + u√m be the fundamental unit of Q(√m), then h∗ ≡ 3tuh if m ≡ 1(4); h∗ ≡ (1 + 2u2)tuh if m ≡ 2(4); 2h∗ = 3tuh if m ≡ 3(8); tuh ≡ 0 if m ≡ 7(8). And if ε = (a + b√m)2, a ≡ b ≡ 1(2), then h∗ ≡ (a ± 1 + (1 − N(ε))ab)h, where 4∣a±1