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

Itamar Gal[1]; Robert Grizzard[2]

• [1] Department of Mathematics, University of Texas at Austin 2515 Speedway, Stop C1200 Austin, TX 78712, U.S.A.
• [2] Department of Mathematics University of Texas at Austin 2515 Speedway, Stop C1200 Austin, TX 78712, U.S.A.
• Volume: 26, Issue: 3, page 655-672
• ISSN: 1246-7405

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## Abstract

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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.

## How to cite

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Gal, Itamar, and Grizzard, Robert. "On the compositum of all degree $d$ extensions of a number field." Journal de Théorie des Nombres de Bordeaux 26.3 (2014): 655-672. <http://eudml.org/doc/275714>.

@article{Gal2014,
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 $d &gt; 2$ 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.},
affiliation = {Department of Mathematics, University of Texas at Austin 2515 Speedway, Stop C1200 Austin, TX 78712, U.S.A.; Department of Mathematics University of Texas at Austin 2515 Speedway, Stop C1200 Austin, TX 78712, U.S.A.},
author = {Gal, Itamar, Grizzard, Robert},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Number fields; infinite algebraic extensions; Galois theory; permutation groups},
language = {eng},
month = {12},
number = {3},
pages = {655-672},
publisher = {Société Arithmétique de Bordeaux},
title = {On the compositum of all degree $d$ extensions of a number field},
url = {http://eudml.org/doc/275714},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Gal, Itamar
AU - Grizzard, Robert
TI - On the compositum of all degree $d$ extensions of a number field
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/12//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 3
SP - 655
EP - 672
AB - 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 $d &gt; 2$ 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.
LA - eng
KW - Number fields; infinite algebraic extensions; Galois theory; permutation groups
UR - http://eudml.org/doc/275714
ER -

## References

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