IJPAM: Volume 112, No. 4 (2017)
GROUPS OVER FIELDS OF CHARACTERISTIC 0 , James Beuerle, Michael Keenan
Department of Mathematics and Statistics
Campus Box 2320, Elon, NC 27244, USA
be an irreducible degree four polynomial defined over a field and let where is a root of in some fixed algebraic closure of . Several methods appear in the literature for computing the Galois group of , most of which rely on forming and factoring resolvent polynomials; i.e., polynomials defining subfields of the splitting field of . This paper surveys those methods that generalize to arbitrary base fields of characteristic 0. Further, we describe a non-resolvent method that determines if has a quadratic subfield by counting the number of roots of that are contained in , 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 is the rational numbers.
Received: July 30, 2016
Revised: February 12, 2016
Published: February 19, 2017
AMS Subject Classification: 12Y05, 20B35, 12F10
Key Words and Phrases: quartic Galois groups, resolvent polynomials, automorphism groups, efficiency
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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Pages: 709 -