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Itamar Gal; Robert Grizzard
On the compositum of all degree $d$ extensions of a number field
Journal de théorie des nombres de Bordeaux, 26 no. 3 (2014), p. 655-672, doi: 10.5802/jtnb.884
Article PDF | Reviews MR 3320497
Class. Math.: 12F10, 11R21, 20B05
Keywords: Number fields, infinite algebraic extensions, Galois theory, permutation groups

Résumé - Abstract

We study the compositum $k^{[d]}$ of all degree $d$ extensions of a number field $k$ in a fixed algebraic closure. We show $k^{[d]}$ contains all subextensions of degree less than $d$ if and only if $d \le 4$. We prove that for $d > 2$ there is no bound $c = c(d)$ on the degree of elements required to generate finite subextensions of $k^{[d]}/k$. Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of $d$, but that one can take $c=d$ when $d$ is prime. This question was inspired by work of Bombieri and Zannier on heights in similar extensions, and previously considered by Checcoli.

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