Let be a primitive 2 m th root of unity. We prove that Z[:]=Z [] if and only if :=n`i for some n, i # Z, i odd. This is the first example of number fields of arbitrarily large degree for which all power bases for the ring of integers are known.
2001 Academic Press
1. Introduction
A number field K is said to have a power basis if its ring of integers is of the form Z[:] for some : # K. It is a well-known problem to determine if a number field has a power basis, and if so, to find all the elements which generate such a basis. The set of generators is stable under integer translation and multiplication by &1; we call : and :$ Z-equivalent if :$=n: for some n # Z. Gyo ry [11] proved that up to Z-equivalence there are only finitely many elements which generate a power basis for any number field K. It is usually very difficult, however, to determine all the generators. For examples of power bases determination in quartic fields see Bremner [1], Kable [13], or Nagell [14]. For examples in prime cyclotomic fields see Bremner [2] or Robertson [15]. Algorithms for finding power bases are developed by Gaa l and Schulte [9] for cubic fields; by Gaa l, Petho , and Pohst (in a series of papers, for example see [6, 7]) for quartic fields; by Gaa l and Gyo ry [5] for quintic fields; and by Gaa l and Pohst [8, 3] for sextic fields. See Gaa l [4] or Gyo ry [10] for a survey of these and other known results on the existence and computation of power bases.
Cyclotomic fields are an interesting case because power bases always exist and in some cases we can find all the generators. In this paper we restrict our attention to 2-power cyclotomic fields. Let be a primitive 2 m th root of unity. It is well known that Z[] is the ring of integers of Q(), and so generates a power basis. Our main theorem (Theorem 1.1) shows that there are no additional non-obvious generators for the ring of integers.