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On a Sequence of Nonsolvable Quintic Polynomials
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Jennifer A. Johnstone and Blair K. Spearman

Mathematics and Statistics

University of British Columbia Okanagan

Kelowna, BC V1V 1V7

Canada

**Abstract:**

Aleksandrov, Kolmogorov and Lavrent'ev state that
*x*^{5} + *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 *x*^{5} + *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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