Hilbert–Speiser number fields for a prime p inside the p-cyclotomic field

Abstract

Let p be a prime number. We say that a number field F satisfies the condition ( H p ′ ) when for any cyclic extension N / F of degree p, the ring O N ′ of p-integers of N has a normal integral basis over O F ′ . It is known that F = Q satisfies ( H p ′ ) for any p. It is also known that when p ⩽ 19 , any subfield F of Q ( ζ p ) satisfies ( H p ′ ) . In this paper, we prove that when p ⩾ 23 , an imaginary subfield F of Q ( ζ p ) satisfies ( H p ′ ) if and only if F = Q ( − p ) and p = 43 , 67 or 163 (under GRH). For a real subfield F of Q ( ζ p ) with F ≠ Q , we give a corresponding but weaker assertion to the effect that it quite rarely satisfies ( H p ′ ) .