## STNB2020(34th edition)

### Kummer theory for number fields via entanglement groups

#### Presenters

Antonella Perucca

#### Abstract

Let $K$ be a number field, and let $G$ be a finitely generated and
torsion-free subgroup of $K^\times$. We are interested in computing the
degree of the cyclotomic-Kummer extension $K(\sqrt[n]{G})$ over $K$, where
$\sqrt[n]{G}$ consists of all $n$-th roots of the elements of $G$. We
approach this problem with the theory of entanglements introduced by
Lenstra, as done by Palenstijn in his PhD thesis for $K=\mathbb Q$. We
develop the theory further and then apply it to compute the above degrees.
This is joint work with Pietro Sgobba and Sebastiano Tronto.

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