Real abelian fields satisfying the Hilbert-Speiser condition for some small primes $p$

Abstract

For a prime number $p$, we say that a number field $F$ satisfies the Hilbert-Speiser condition $(H_{p})$ if each tame cyclic extension $N/F$ of degree $p$ has a normal integral basis. In this note, we determine the real abelian number fields satisfying $(H_{p})$ for odd prime numbers $p$ with $h(\mathbf{Q}(\sqrt{-p}))=1$.