Universal normal bases for the abelian closure of the field of rational numbers

1. Introduction. If E/F is a finite-dimensional Galois extension with Galois group G, then, by the Normal Basis Theorem, there exist elements w ∈ E such that {g(w) | g ∈ G} is an F-basis of E, a so-called normal basis, whence w is called normal in E/F. In the present paper, we study normal bases for cyclotomic fields. Let Q be the field of rational numbers; for a positive integer n, we let Q n denote the nth cyclotomic field, i.e., Q n = Q(ζ n), where ζ n is a primitive nth root of unity. For the basics on cyclotomic fields, we refer to [Ri] or [Wa]; we just remark that Q n = Q m with n > m holds if and only if n = 2m and m is odd. As index set for the cyclotomic fields we therefore use the set N of positive integers which are either odd or divisible by 4. Thus, if n, e ∈ N , then Q n ⊆ Q e if and only if n divides e. We call Q e /Q n a cyclotomic extension.