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Abstract

Let $p$ be a rational prime, $G$ a group of order $p$, and $K$ a number field containing a primitive $p$th root of unity. We show that every tamely ramified Galois extension of $K$ with Galois group isomorphic to $G$ has a normal integral basis if and only if for every Galois extension $L/K$ with Galois group isomorphic to $G$, the ring of integers $O_L$ in $L$ is free as a module over the associated order ${\cal A}_{L/K}$. We also give examples, some of which show that this result can still hold without the assumption that $K$ contains a primitive $p$th root of unity.