Antonella Perucca [a]
The degree of non-Galois Kummer extensions of number fields
Pages: 301-313
Received: 5 July 2020
Accepted in revised form: 27 August 2020
Mathematics Subject Classification (2010): Primary: 11Y40; Secondary: 11R20, 11R21.
Keywords: Number field, Kummer theory, Kummer extension, degree.
Author address:
[a]: Department of Mathematics, University of Luxembourg, 6 av. de la Fonte, 4364 Esch-sur-Alzette, Luxembourg
Abstract: Let \(K\) be a number field, and let \(a_1,\ldots, a_r\) be elements of \(K^\times\) which generate a torsion-free subgroup of \(K^\times\) of positive rank \(r\). Let \(\alpha_1,\ldots, \alpha_r\) be \(\ell^n\)-th roots of the given elements respectively, where \(\ell\) is a prime number and \(n> 0\). In this short note we provide explicit parametric formulas for the degree of the extension \(K({\alpha_1},\ldots, {\alpha_r})\). This degree may depend on the choice of \(\alpha_1,\ldots, \alpha_r\) because we are not assuming that the \(\ell^n\)-th roots of unity are in \(K\).