arXiv:0806.0258v1 [math.NT] 2 Jun 2008 ON THE EQUIVALENCE OF THE RESTRICTED HILBERT-SPEISER AND LEOPOLDT PROPERTIES JAMES E. CARTER, CORNELIUS GREITHER, AND HENRI JOHNSTON Abstract. Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K, the ring of integers O L is free as an O K G-module. If O L is free over the associated order A L/K for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-Speiser of type G. We show that the converse holds for many number fields K (in particular, for K/Q Galois) when G = C p has prime order. In the process, we prove that if p ≥ 7 (or p = 5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilbert-Speiser of type C p . 1. Introduction Let L/K be a finite abelian extension of number fields with Galois group G. The associated order is defined to be A L/K := {x ∈ KG : x(O L ) ⊆O L }. In the case K = Q, Leopoldt’s Theorem [10] shows that the ring of integers O L of L is free as a module over A L/Q . (A simplified proof of this result can be found in [11].) More generally, we say that a number field K is Leopoldt if for every finite abelian extension L/K , the ring of integers O L is free over A L/K (note that this differs from the definition of Leopoldt given in [8]). Since A L/K = O K G if and only if L/K is tame, Leopoldt’s Theorem implies the celebrated Hilbert- Speiser Theorem: Every tame finite abelian extension of Q has a normal integral basis, that is, O L is free as a ZG-module. (In this paper, we shall take “tame” to mean “at most tamely ramified”.) A number field K is called a Hilbert-Speiser field if for every finite abelian tame extension L/K , the ring of integers O L is free over O K G; in particular, Q is such a field. The same reasoning as above shows that if K is Leopoldt then K is Hilbert-Speiser. The converse follows from Leopoldt’s Theorem and the result proven in [6] that Q is the only Hilbert-Speiser field. Hence we have the following observation. Theorem 1.1. Let K be a number field. Then K is a Hilbert-Speiser field if and only if K is a Leopoldt field. The question arises as to whether a similar result holds when one fixes the group G. Date : 1st June, 2008. 2000 Mathematics Subject Classification. Primary 11R33. Key words and phrases. Galois module structure, normal integral basis, associated order, Hilbert-Speiser field, Leopoldt field. Johnston was partially supported by a grant from the Deutscher Akademischer Austausch Dienst. 1