Missouri Journal of Mathematical Sciences

Centralizers of Transitive Permutation Groups and Applications to Galois Theory

Chad Awtrey, Nakhila Mistry, and Nicole Soltz

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Abstract

Let $f(x)$ be an irreducible polynomial of degree $n$ defined over a field $F$, and let $G$ be the Galois group of $f$, identified as a transitive subgroup of $S_n$. Let $K/F$ be the stem field of $f$. We show the automorphism group of $K/F$ is isomorphic to the centralizer of $G$ in $S_n$. We include two applications to computing Galois groups; one in the case $F$ is the rational numbers, the other when $F$ is the 5-adic numbers.

Article information

Source
Missouri J. Math. Sci., Volume 27, Issue 1 (2015), 16-32.

Dates
First available in Project Euclid: 3 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1449161364

Digital Object Identifier
doi:10.35834/mjms/1449161364

Mathematical Reviews number (MathSciNet)
MR3431112

Zentralblatt MATH identifier
1332.12005

Subjects
Primary: 20B35: Subgroups of symmetric groups
Secondary: 12Y05: Computational aspects of field theory and polynomials 11R32: Galois theory 11S20: Galois theory

Keywords
Galois group computation centalizers normalizers $p$-adic fields quartic extensions

Citation

Awtrey, Chad; Mistry, Nakhila; Soltz, Nicole. Centralizers of Transitive Permutation Groups and Applications to Galois Theory. Missouri J. Math. Sci. 27 (2015), no. 1, 16--32. doi:10.35834/mjms/1449161364. https://projecteuclid.org/euclid.mjms/1449161364


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