STNB2020(34th edition)

Kummer theory for number fields via entanglement groups


Antonella Perucca


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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