Congruences for class numbers of Q(√±2p) when p≡3 (mod 4) is prime

For a prime p equivalent to 3 (mod 4),let h(-8p) and h(8p) be the class numbers of Q(root-2p) and Q(root 2p), respectively. Let(xi)be the Hirzebruch sum of a quadratic irrational xi. We show that h(-8p)equivalent to h(8p)((2 root 2p)/3-(1+root 2p2)/3)(mod 16).Also, we show that h(-8p)equivalent to 2h(8p)(2 root 2p)/3(mod 8)ifp equivalent to 3(mod 8), and h(-8p)equivalent to(2h(8p)(2 root 2p)/3)+4(mod 8)if p equivalent to 7(mod 8).