On the compositum of all degree $d$ extensions of a number field

Gal, Itamar; Grizzard, Robert

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 655-672.

Nous étudions le compositum $k^{\left[d\right]}$ de toutes les extensions de degré $d$ d’un corps de nombres $k$ dans une clôture algébrique fixée. Nous démontrons que $k^{\left[d\right]}$ contient toutes les sous-extensions de degré inférieur à $d$ si et seulement si $d\le 4$. Nous montrons que quand $d>2$, il n’existe pas de majorant $c=c\left(d\right)$ sur le degré des éléments nécessaires pour engendrer les sous-extensions finies de $k^{\left[d\right]}/k$. En se restreignant aux sous-extensions galoisiennes, nous montrons qu’un tel majorant n’existe pas sous certaines conditions sur les diviseurs de $d$, mais que l’on peut prendre $c=d$ quand $d$ est premier. Cette question a été inspirée par les travaux de Bombieri et Zannier sur les hauteurs dans des extensions similaires, et examinés par Checcoli.

We study the compositum $k^{\left[d\right]}$ of all degree $d$ extensions of a number field $k$ in a fixed algebraic closure. We show $k^{\left[d\right]}$ 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 $c=c\left(d\right)$ on the degree of elements required to generate finite subextensions of $k^{\left[d\right]}/k$. 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.

Classification : 12F10, 11R21, 20B05
Mots clés : Number fields, infinite algebraic extensions, Galois theory, permutation groups

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