Kuroda's class number formula

Abstract

Let k be a number field and K/k a V4-extension, i.e., a normal extension with Gal(K/k) = V4, where V4 is Klein’s four-group. K/k has three intermediate fields, say k1, k2, and k3. We will use the symbol N i (resp. Ni) to denote the norm of K/ki (resp. ki/k), and by a widespread abuse of notation we will apply N i and Ni not only to numbers, but also to ideals and ideal classes. The unit groups (groups of roots of unity, , groups of fractional ideals, class numbers) in these fields will be denoted by Ek, E1, E2, E3, EK (Wk,W1, . . . , JK , J1, . . . , hk, h1, . . . ) respectively, and the (finite) index q(K) = EK : E1E2E3) is called the unit index of K/k. For k = Q, k1 = Q( √ −1 ) and k2 = Q( √ m ) it was already known to Dirichlet [5] that hK = 12q(K)h2h3. Bachmann [2], Amberg [1] and Herglotz [12] generalized this class number formula gradually to arbitrary extensions K/Q whose Galois groups are elementary abelian 2-groups. A remark of Hasse [11, p. 3] seems to suggest that Varmon [30] proved a class number formula for extensions with Gal(K/k) an elementary abelian p-group; unfortunately, his paper was not accessible to me. Kuroda [18] later gave a formula in case there is no ramification at the infinite primes. Wada [31] stated a formula for 2-extensions of k = Q without any restriction on the ramification (and without proof), and finally Walter [32] used Brauer’s class number relations to deduce the most general Kuroda-type formula. As we shall see below, Walter’s formula for V4-extensions does not always give correct results if K contains the 8th roots of unity. This does not, however, seem to effect the validity of the work of Parry [22, 23] and Castela [4], both of whom made use of Walter’s formula. The proofs mentioned above use analytic methods; for V4-extensions K/Q, however, there exist algebraic proofs given by Hilbert [14] (if √ −1 ∈ K), Kuroda [17] (if √ −1 ∈ K), Halter-Koch [9] (if K is imaginary), and Kubota [15, 16]. For base fields k 6= Q, on the other hand, no non-analytic proofs seem to be known except for very special cases (see e.g. the very recent work of Berger [3]). In this paper we will show how Kubota’s proof can be generalized. The proof consists of two parts; in the first part, where we measure the extent to which Cl(K) is generated by classes coming from the Cl(ki), we will use class field theory in its ideal-theoretic formulation (see Hasse [10] or Garbanati [7]). The second part of the proof is a somewhat lengthy index computation.