Divisibility of class numbers: enumerative approach

It is well-known since Gauss that infinitely many quadratic fields have even class number. In fact, if K is a quadratic field of discriminant D, having r prime divisors, then the class number hK is divisible by 2 if D < 0 and by 2 if D > 0. See [4, Theorem 3.8.8] for a more precise statement. In 1922 Nagell [17, Satz VI] obtained the following remarkable result: given a positive integer l, there exist infinitely many imaginary quadratic fields with class number divisible by l. See [2] for a different proof. It took almost half a century to extend Nagell’s result to real quadratic field, see Yamamoto [29] and Weinberger [28]. Uchida [27] extended this to cyclic cubic fields. In mid-eighties, Azuhata and Ichimura [3] and Nakano [18, 19] obtained similar results for fields of arbitrary degree n. Recently Murty [16] gave quantitative versions of the theorems of Nagell and YamamotoWeinberger on quadratic fields. He proved that for all sufficiently large X there exist at least c(l)X imaginary quadratic fields and at least c(l, ε)X real quadratic fields with class number divisible by l and discriminant not exceeding X in absolute value. (The second exponent can be replaced by 1/2l− ε if l is odd.) Various refinement and extensions of Murty’s results were suggested in [5, 15, 25, 30]. Much less is known for fields of higher degree. In [13], it is shown that at least c(l)X/ logX pure cubic fields have discriminant not exceeding X and class number divisible by l. In this paper, we extend these results to fields of degree n ≥ 3. Theorem 1.1 Let n and l be positive integers, n ≥ 3, and put μ = 1 2(n−1)l . There exist positive real numbers X0 = X0(n, l) and c = c(n, l) with the following property. For any X > X0 there is at least cX pairwise non-isomorphic number fields of degree n, discriminant not exceeding X , and the class number divisible by l.