# 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.

Journal de Théorie des Nombres de Bordeaux (2014)

- Volume: 26, Issue: 3, page 655-672
- ISSN: 1246-7405

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topGal, 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 $d > 2$ 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 $d > 2$ 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 -

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