The intersection of cyclic Kummer extensions with cyclotomic extensions

Details The intersection of cyclic Kummer extensions with cyclotomic extensions The intersection of cyclic Kummer extensions with cyclotomic extensions

2011-07-22

arxiv.org/abs/1107.4595v3

Mathematics

Number Theory

v3: revised expanded appendix; added alternative proof of Lorenz's result; 20 pages

Scientific paper

Let $K$ be a field, and $\ell$ a prime number different from the characteristic of $K$. Consider an extension of $K$ of the form $K(\zeta_{\ell^m}, \sqrt[\ell^n]{a})$, obtained by adding a root of unity of order $\ell^m$ and the $\ell^n$-th root of some element of $K^\times$, with $m\geq n$. We give a formula for the degree of this extension, which depends only on few parameters. As an application, if $K$ is a number field we calculate the density of primes $\mathfrak p$ of $K$ such that the order of $(a \bmod \mathfrak p)$ is coprime to $\ell$. This work is based on a result by Schinzel of 1977 describing abelian radical extensions.

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