The irreducibility of power compositional sextic polynomials and their Galois groups
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Joshua Harrington [1] ; Lenny Jones [2]
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[1] Cedar Crest College-USA
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[2] Shippensburg University-USA
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- Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 120, Nº 2, 2017, págs. 181-194
- Idioma: inglés
- DOI: 10.7146/math.scand.a-25850
- Texto completo no disponible (Saber más ...)
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Resumen
We define a power compositional sextic polynomial to be a monic sextic polynomial f(x):=h(xd)∈Z[x], where h(x) is an irreducible quadratic or cubic polynomial, and d=3 or d=2, respectively. In this article, we use a theorem of Capelli to give necessary and sufficient conditions for the reducibility of f(x), and also a description of the factorization of f(x) into irreducibles when f(x) is reducible. In certain situations, when f(x) is irreducible, we also give a simple algorithm to determine the Galois group of f(x) without the calculation of resolvents. The algorithm requires only the use of the Rational Root Test and the calculation of a single discriminant. In addition, in each of these situations, we give infinite families of polynomials having the possible Galois groups.
