The class number of $Q(\sqrt{-2p}) modulo 8, for $p\equiv5 \mod 8$ is prime
Kenneth S. Williams · Rocky Mountain Journal of Mathematics · 1981
Let/? = 5 (mod 8) be a prime.Let h(± 2p) denote^the class number of the quadratic field Q(V±2p).Let T + U V2p be the fundamental unit of Q(*/2p).It is shown that h{ -2p) = h{2p) + 2T +2 (mod 8).If p is a prime congruent to 5 modulo 8, it is well known that the class number h{ -2p) of the imaginary quadratic field Q{\/ -2p) is congruent to 2 modulo 4 (see for example [2: p. 413]).In this paper, we determine h{ -2p) modulo 8.This is a problem of D.H. Lehmer [6: p. 10].(The corresponding problem for h(-p), p = 3 (mod 4), has been solved by the author in [9].)We let e 2p = T 4-U^2p be the fundamental unit of the real quadratic field Q(\/2p), so that rand £/are positive integers.It is a classical theorem of Dirichlet [5: p. 226] that e 2p has norm -1, that is, N(e 2p ) = T*-2pU2= -1, from which it follows that T and U are both odd.The class number h{2p) of Q(^/2p) is also congruent to 2 modulo 4 (see for example [3: p. 101]).With this notation we prove the following theorem.THEOREM.h{-2p) = h(2p) + 2T + 2 (mod 8).PROOF.It is assumed throughout that p is a prime congruent to 5 modulo 8.We set p = exp(27r///?).For z a complex variable, we let (1)where F(z) is the cyclotomic polynomial of index /?, MOS Subject Classification Numbers: 12A25,12A50.