Wild primes of a self-equivalence of a number field

Alfred Czogała¹ ; Beata Rothkegel¹

¹ Institute of Mathematics University of Silesia Bankowa 14 40-007 Katowice, Poland

Acta Arithmetica, Tome 166 (2014) no. 4, pp. 335-348

Résumé

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 $K$.

Détail

DOI : 10.4064/aa166-4-2

Keywords: number field assume rank ideal class group equal rank narrow ideal class group moreover assume has unique dyadic prime mathfrak class mathfrak square ideal class group prove mathfrak dots mathfrak finite primes bullet class mathfrak square ideal class group every dots bullet local square mathfrak every nondyadic mathfrak mathfrak dots mathfrak mathfrak dots mathfrak wild set self equivalence field