Summary: Aleksandrov, Kolmogorov and Lavrent'ev state that $x^{5} + x - a$ is nonsolvable for $a = 3,4,5$,7,8,9,10,11,$\dots $. 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.

12E05, 14G05

nonsolvable polynomial, Galois group