Polynomials over Q solving an embedding problem

Nuria Vila

Annales de l'institut Fourier (1985)

  • Volume: 35, Issue: 2, page 79-82
  • ISSN: 0373-0956

Abstract

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The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group A n , can be embedded in any central extension of A n if and only if n ≡ 0 ( m o d 8 ) , or n ≡ 2 ( m o d 8 ) and n is a sum of two squares. Consequently, for theses values of n , every central extension of A n occurs as a Galois group over Q .

How to cite

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Vila, Nuria. "Polynomials over $Q$ solving an embedding problem." Annales de l'institut Fourier 35.2 (1985): 79-82. <http://eudml.org/doc/74678>.

@article{Vila1985,
abstract = {The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group $A_ n$, can be embedded in any central extension of $A_ n$ if and only if $n\equiv 0 (mod 8)$, or $n\equiv 2 (mod 8)$ and $n$ is a sum of two squares. Consequently, for theses values of $n$, every central extension of $A_ n$ occurs as a Galois group over $\{\bf Q\}$.},
author = {Vila, Nuria},
journal = {Annales de l'institut Fourier},
keywords = {embedding in central extension of ; Galois group of equation; Galois extension with alternating group; central extension of as Galois group; polynomials},
language = {eng},
number = {2},
pages = {79-82},
publisher = {Association des Annales de l'Institut Fourier},
title = {Polynomials over $Q$ solving an embedding problem},
url = {http://eudml.org/doc/74678},
volume = {35},
year = {1985},
}

TY - JOUR
AU - Vila, Nuria
TI - Polynomials over $Q$ solving an embedding problem
JO - Annales de l'institut Fourier
PY - 1985
PB - Association des Annales de l'Institut Fourier
VL - 35
IS - 2
SP - 79
EP - 82
AB - The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group $A_ n$, can be embedded in any central extension of $A_ n$ if and only if $n\equiv 0 (mod 8)$, or $n\equiv 2 (mod 8)$ and $n$ is a sum of two squares. Consequently, for theses values of $n$, every central extension of $A_ n$ occurs as a Galois group over ${\bf Q}$.
LA - eng
KW - embedding in central extension of ; Galois group of equation; Galois extension with alternating group; central extension of as Galois group; polynomials
UR - http://eudml.org/doc/74678
ER -

References

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  1. [1] B. HUPPERT, Endliche Gruppen I, Die Grund. der Math. Wiss., 134, Springer, 1967. Zbl0217.07201MR37 #302
  2. [2] E. NART and N. VILA, Equations with absolute Galois group isomorphic to An, J. Number Th., 16 (1983), 6-13. Zbl0511.12010MR85b:11081
  3. [3] I. SCHUR, Ùber die Darstellungen der symmetrischen und alternierender Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math., 139 (1911), 155-250. Zbl42.0154.02JFM42.0154.02
  4. [4] J.-P. SERRE, L'invariant de Witt de la forme Tr (x2), Com. Math. Helv., to appear. Zbl0565.12014
  5. [5] N. VILA, On central extensions of An as Galois group over Q, to appear. Zbl0562.12011

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