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AN INFINITE FAMILY OF NINTH DEGREE DIHEDRAL POLYNOMIALS

Part of: Algebraic number theory: global fields Field extensions

Published online by Cambridge University Press:  14 August 2017

LENNY JONES Open the ORCID record for LENNY JONES [Opens in a new window]  and
TRISTAN PHILLIPS Open the ORCID record for TRISTAN PHILLIPS [Opens in a new window]
LENNY JONES*
Affiliation:
Department of Mathematics, Shippensburg University, Shippensburg, PA 17257, USA email [email protected]
TRISTAN PHILLIPS
Affiliation:
Department of Mathematics, Shippensburg University, Shippensburg, PA 17257, USA email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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For any integer $m\neq 0$, we prove that $f(x)=x^{9}+9mx^{6}+192m^{3}$ is irreducible over $\mathbb{Q}$ and that the Galois group of $f(x)$ over $\mathbb{Q}$ is the dihedral group of order 18. Moreover, we show that for infinitely many values of $m$, the splitting fields for $f(x)$ are distinct.

Keywords

dihedral groupGalois groupirreducible polynomialCapelli

MSC classification

Primary: 12F10: Separable extensions, Galois theory
Secondary: 11R09: Polynomials (irreducibility, etc.) 11R32: Galois theory 12F12: Inverse Galois theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 1 , February 2018 , pp. 47 - 53
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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