ON A THEOREM OF CHILDS ON NORMAL BASES OF RINGS OF INTEGERS

Abstract

Let p be a prime number, F a number field, and H n the set of all unramified cyclic extensions over F of degree p having a relative normal integral basis. When ζp ∈ F×, Childs determined the set H n in terms of Kummer generators. When p=3 and F is an imaginary quadratic field, Brinkhuis determined this set in a form which is, in a sense, analogous to Childs's result. The paper determines this set for all p ⩾ 3 and F with ζp ∉ F× (and satisfying an additional condition), using the result of Childs and a technique developed by Brinkhuis. Two applications are also given.