On the unramified common divisor of discriminants of integers in a normal extension

Abstract

Let $F$ be an algebraic number field of a finite degree, and $K$ be a normal extension over $F$ of a finite degree $n$. Let $\mathfrak{p}$ be a prime ideal of $F$ which is unramified in $K/F$, $\mathfrak{P}$ be a prime ideal of $K$ dividing $\mathfrak{p}$ such that $N_{K/F}\mathfrak{P} = \mathfrak{p}^f$, $n=fg$. Denote by $\delta(K/F)$ the greatest common divisor of discriminants of integers of $K$ with respect to $K/F$. Then, $\mathfrak{p}$divides $\delta(K/F)$ if and only if $\Sigma_{d|f} \mu(\frac fd)N\mathfrak{p}^d < n$.