Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.8

On a Sequence of Nonsolvable Quintic Polynomials

Jennifer A. Johnstone and Blair K. Spearman
Mathematics and Statistics
University of British Columbia Okanagan
Kelowna, BC V1V 1V7


Aleksandrov, Kolmogorov and Lavrent'ev state that x5 + x - a is nonsolvable for a = 3,4,5,7,8,9,10,11,... . In other words, these polynomials have a nonsolvable Galois group. A full explanation of this sequence requires consideration of both reducible and irreducible solvable quintic polynomials of the form x5 + x - a. All omissions from this sequence due to solvability are characterized. This requires the determination of the rational points on a genus 3 curve.

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Received November 11 2008; revised version received February 13 2009. Published in Journal of Integer Sequences, February 15 2009.

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