Ideal Class Groups of Cyclotomic Number Fields Ii

We first study some families of maximal real subfields of cyclotomic fields with even class number, and then explore the implications of large plus class numbers of cyclotomic fields. We also discuss capitulation of the minus part and the behaviour of p-class groups in cyclic ramified p-extensions. This is a continuation of [13]; parts I and II are independent, but will be used in part III. 6. The 2-Class Group Let h(m) and h(m) denote the class number of Q(ζm) and Q(ζm+ ζ−1 m ), respectively, and put h−(m) = h(m)/h(m). In this section we will show how results on the 2-class field tower of quadratic number fields can be used to improve results of Stevenhagen [30] on the parity of h(m) for certain composite m with few prime factors. Proposition 4. Let p ≡ q ≡ 1 mod 4 be primes, put L = Q(ζpq), and let K and K be the maximal 2-extensions contained in L and L = L ∩ R, respectively. (1) 2 | h(K) if and only if (p/q) = 1; (2) if (p/q) = 1 and (p/q)4 = (q/p)4, then 2 | h(F ) for every subfield F ⊆ L containing Q(√pq ); (3) if (p/q)4 = (q/p)4 = +1, then 4 | h(K). Proof. By a result of Rédei and Reichardt [23, 24], the quadratic number field k = Q(√pq ) admits a cyclic quartic extension F/k which is unramified outside ∞ and which is normal over Q with Gal (F/Q) ' D4, the dihedral group of order 8. The last property guarantees that F is either totally real or totally complex; Scholz [26] has shown that F is real if and only if (p/q)4 = (q/p)4. Assume that F is real; then, for every subfield M of K containing Q(√pq ), the extension FK/K is unramified everywhere and is cyclic 1991 Mathematics Subject Classification. Primary 11 R 21; Secondary 11 R 29, 11 R 18.