We study the compositum of all degree extensions of a number field in a fixed algebraic closure. We show contains all subextensions of degree less than if and only if . We prove that for there is no bound on the degree of elements required to generate finite subextensions of . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of , but that one can take when is prime. This question was inspired by work of Bombieri and Zannier on heights in similar extensions, and previously considered by Checcoli.
Nous étudions le compositum de toutes les extensions de degré d’un corps de nombres dans une clôture algébrique fixée. Nous démontrons que contient toutes les sous-extensions de degré inférieur à si et seulement si . Nous montrons que quand , il n’existe pas de majorant sur le degré des éléments nécessaires pour engendrer les sous-extensions finies de . En se restreignant aux sous-extensions galoisiennes, nous montrons qu’un tel majorant n’existe pas sous certaines conditions sur les diviseurs de , mais que l’on peut prendre quand 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.
Keywords: Number fields, infinite algebraic extensions, Galois theory, permutation groups
Itamar Gal 1; Robert Grizzard 2
@article{JTNB_2014__26_3_655_0, author = {Itamar Gal and Robert Grizzard}, title = {On the compositum of all degree $d$ extensions of a number field}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {655--672}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {26}, number = {3}, year = {2014}, doi = {10.5802/jtnb.884}, zbl = {06561053}, mrnumber = {3320497}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.884/} }
TY - JOUR AU - Itamar Gal AU - Robert Grizzard TI - On the compositum of all degree $d$ extensions of a number field JO - Journal de théorie des nombres de Bordeaux PY - 2014 SP - 655 EP - 672 VL - 26 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.884/ DO - 10.5802/jtnb.884 LA - en ID - JTNB_2014__26_3_655_0 ER -
%0 Journal Article %A Itamar Gal %A Robert Grizzard %T On the compositum of all degree $d$ extensions of a number field %J Journal de théorie des nombres de Bordeaux %D 2014 %P 655-672 %V 26 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.884/ %R 10.5802/jtnb.884 %G en %F JTNB_2014__26_3_655_0
Itamar Gal; Robert Grizzard. On the compositum of all degree $d$ extensions of a number field. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 655-672. doi : 10.5802/jtnb.884. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.884/
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