Abstract
We describe two new algorithms for determining the Galois group of an irreducible quintic polynomial f defined over a field F. For one approach, we introduce a single degree 24 resolvent polynomial construction, the degrees of whose irreducible factors completely determine the Galois group of f. Our other approach, which does not rely on factoring resolvent polynomials of degree greater than two, considers the discriminant of f along with the size of the automorphism group of the polynomial’s stem field. We show that this second method is particularly effective in the case where F is a finite extension of the p-adic numbers.
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Acknowledgements
The authors would like to thank the anonymous reviewer for the careful reading and helpful comments. The authors would also like to thank Elon University for supporting this project through internal grants and the Center for Undergraduate Research in Mathematics for their grant support. This research was supported in part by NSF grant # DMS-1148695.
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Awtrey, C., Shill, C.R. (2015). Absolute Resolvents and Masses of Irreducible Quintic Polynomials. In: Rychtář, J., Chhetri, M., Gupta, S., Shivaji, R. (eds) Collaborative Mathematics and Statistics Research. Springer Proceedings in Mathematics & Statistics, vol 109. Springer, Cham. https://doi.org/10.1007/978-3-319-11125-4_4
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DOI: https://doi.org/10.1007/978-3-319-11125-4_4
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