IJPAM: Volume 112, No. 4 (2017)

Title

ALGORITHMS FOR COMPUTING QUARTIC GALOIS
GROUPS OVER FIELDS OF CHARACTERISTIC 0

Authors

Chad Awtrey$^1$, James Beuerle$^2$, Michael Keenan$^3$
$^{1,2,3}$Department of Mathematics and Statistics
Elon University
Campus Box 2320, Elon, NC 27244, USA

Abstract

Let $f(x)$ be an irreducible degree four polynomial defined over a field $F$ and let $K=F(\alpha)$ where $\alpha$ is a root of $f$ in some fixed algebraic closure $\overline{F}$ of $F$. Several methods appear in the literature for computing the Galois group $G$ of $f$, most of which rely on forming and factoring resolvent polynomials; i.e., polynomials defining subfields of the splitting field of $f$. This paper surveys those methods that generalize to arbitrary base fields of characteristic 0. Further, we describe a non-resolvent method that determines if $K$ has a quadratic subfield by counting the number of roots of $f$ that are contained in $K$, and we also describe how to construct explicitly a polynomial defining a quadratic subfield. We end with a comparison of run times for the various algorithms in the case $F$ is the rational numbers.

History

Received: July 30, 2016
Revised: February 12, 2016
Published: February 19, 2017

AMS Classification, Key Words

AMS Subject Classification: 12Y05, 20B35, 12F10
Key Words and Phrases: quartic Galois groups, resolvent polynomials, automorphism groups, efficiency

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Bibliography

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How to Cite?

DOI: 10.12732/ijpam.v112i4.5 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 112
Issue: 4
Pages: 709 -


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