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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. 

References

Beachy, J. A. and Blair, W. D., Abstract Algebra, 3rd edn (Waveland Press, Inc., Long Grove, IL, 2005).Google Scholar
Brown, S., Spearman, B. and Yang, Q., ‘On sextic trinomials with Galois group C 6 , S 3 or C 3 × S 3 ’, J. Algebra Appl. 12(1) (2013), Article ID 1250128, 9 pages.Google Scholar
Butler, G. and McKay, J., ‘The transitive groups of degree up to eleven’, Comm. Algebra 11(8) (1983), 863–911.Google Scholar
Cohen, H., A Course in Computational Algebraic Number Theory (Springer, Berlin, 2000).Google Scholar
Harrington, J. and Jones, L., ‘The irreducibility of power compositional sextic polynomials and their Galois groups’, Math. Scand. 120(2) (2017), 181–194.Google Scholar
Ide, J. and Jones, Lenny, ‘Infinite families of A 4 -sextic polynomials’, Canad. Math. Bull. 57(3) (2014), 538–545.CrossRefGoogle Scholar
Jensen, C. U., Ledet, A. and Yui, N., ‘Generic polynomials’, in: Constructive Aspects of the Inverse Galois Problem, Mathematical Sciences Research Institute Publications, 45 (Cambridge University Press, Cambridge, 2002).Google Scholar
Schinzel, A., Polynomials with Special Regard to Reducibility, Encyclopedia of Mathematics and its Applications, 77 (Cambridge University Press, Cambridge, 2000).Google Scholar
Spearman, B. and Williams, K., ‘Quartic trinomials with Galois groups A 4 and V 4 ’, Far East J. Math. Sci. 2(5) (2000), 665–672.Google Scholar
Spearman, B. and Williams, K., ‘The simplest D 4 -octics’, Int. J. Algebra 2(1–4) (2008), 79–89.Google Scholar
Williamson, C. J., ‘Odd degree polynomials with dihedral Galois groups’, J. Number Theory 34(2) (1990), 153–173.CrossRefGoogle Scholar