On totally real Hilbert-Speiser Fields of type C_p

TL;DR

This paper investigates conditions under which totally real number fields are not Hilbert-Speiser fields of type C_p, especially focusing on ramification at prime p and the cases where p is at least 5.

Contribution

It establishes new non-existence results for totally real Hilbert-Speiser fields of type C_p when p ≥ 7 or p=5 with additional conditions, advancing understanding of normal integral bases.

Findings

Totally real fields ramified at p are not Hilbert-Speiser of type C_p for p ≥ 7.

For p=5, additional conditions determine non-Hilbert-Speiser status.

Provides criteria linking ramification and the existence of normal integral bases.

Abstract

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers O_L is free as an O_K[G]-module. Let C_p denote the cyclic group of prime order p. We show that if p >= 7 (or p=5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilbert-Speiser of type C_p.