Wild primes of a self-equivalence of a number field
Volume 166 / 2014
Acta Arithmetica 166 (2014), 335-348
MSC: Primary 11E12; Secondary 11E81.
DOI: 10.4064/aa166-4-2
Abstract
Let $K$ be a number field. Assume that the 2-rank of the ideal class group of $K$ is equal to the 2-rank of the narrow ideal class group of $K$. Moreover, assume $K$ has a unique dyadic prime $\mathfrak d$ and the class of $\mathfrak d$ is a square in the ideal class group of $K$. We prove that if $\mathfrak p_1,\dots,\mathfrak p_n$ are finite primes of $K$ such that
$\bullet$ the class of $\mathfrak p_i$ is a square in the ideal class group of $K$ for every $i\in\{1,\dots,n\}$,
$\bullet$ $-1$ is a local square at $\mathfrak p_i$ for every nondyadic $\mathfrak p_i\in\{\mathfrak p_1,\dots,\mathfrak p_n\}$,then $\{\mathfrak p_1,\dots,\mathfrak p_n\}$ is the wild set of some self-equivalence of the field $K$.
