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