Unimodular integral circulants

Abstract

Some properties of the "discriminant matrix" (α_i^(S_k))) of a normal algebraic number field of degree n were investigated in two previous notes (1 ,2). Here the α_i form an integral basis of the field and the S_k are the elements of the GALOIS group). In the special case when the α_i form a normal basis various problems concerning group matrices arise, among others, questions concerning unimodular group matrices whose elements are rational integers. If the field is cyclic circulant, matrices appear, i.e. matrices C = (c_(ik)) with c_(ik) = c_(k-i+1) where the suffixes are considered mod n. In particular the following theorem was obtained which will be studied further in the present note.

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