Appendix To: Dirichlet Series Associated to Quartic Fields with given Resolvent

This is an appendix to our paper [1], where we give an explicit formula for the Dirichlet series ∑ K |Disc(K)|−s, where the sum is over isomorphism classes of all quartic fields whose cubic resolvent field is isomorphic to k. In the present note, we give a complete proof of a theorem enumerating splitting types of certain number fields, which was stated in [1] without a complete proof. The details are somewhat long and not terribly difficult, and so we decided to leave them out of [1]. The results proved here were largely (and independently) also obtained by Martinet [2], again in unpublished work. This is an appendix to [1], not intended for publication. Accordingly we refer to [1] for the motivation for proving Theorem 0.4. Here we simply commence with the details, although this note should be fairly readable on its own. We also note that many of the results in this paper were obtained independently and previously in unpublished work of Martinet [2]. In [1] and the present note, we are interested in studying A4and S4-quartic fields; i.e., quartic fields whose Galois closure is isomorphic to A4 or S4 respectively. In the A4 case, K̃ contains a unique cyclic cubic subfield k, and in the S4 case, K̃ contains three isomorphic noncyclic cubic subfields k. In either case k is called the cubic resolvent of K, it is unique up to isomorphism, and it satisfies Disc(K) = Disc(k)f(K) for some integer f(K). Definition 0.1. Given any cubic field k (cyclic or not), let L(k) be the set of isomorphism classes of quartic fields whose resolvent cubic is isomorphic to k, with the additional restriction that the quartic is totally real when k is such. Furthermore, for any n define L(k, n) to be the subset of L(k) of those fields with discriminant equal to nDisc(k). Finally, we define Ltr(k, 64) to be the subset of those L ∈ L(k, 64) such that 2 is totally ramified in L, and we set L2(k) = L(k, 1) ∪ L(k, 4) ∪ L(k, 16) ∪ Ltr(k, 64) . Note that if k is totally real the elements of F(k) are totally real or totally complex, and L(k) is the subset of totally real ones, while if k is complex then the elements of L(k) = F(k) have mixed signature r1 = 2, r2 = 1. We introduce some standard notation for splitting types of primes in a number field. If L is, say, a quartic field, and p is a prime for which (p) = p1p2 in L, where pi has residue class degree i for i = 1, 2, we say that p has splitting type (21) in L (or simply that p is (21) in L). Other splitting types such as (22), (1111), (1), etc. are defined similarly. Moreover, when 2 has type (11) in a cubic field k, we say that 2 has type (11)0 or (1 1)4 depending on whether Disc(k) ≡ 0 (mod 8) or Disc(k) ≡ 4 (mod 8).