On a Sequence of Nonsolvable Quintic Polynomials
Jennifer A. Johnstone, Blair Kenneth Spearman · 2009
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. 1