Geodesic
On a sequence of nonsolvable quintic polynomials
Journal of integer sequences, Tome 12 (2009) no. 2.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 12E05, 14G05
Keywords: nonsolvable polynomial, Galois group 
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Johnstone, Jennifer A.; Spearman, Blair K. On a sequence of nonsolvable quintic polynomials. Journal of integer sequences, Tome 12 (2009) no. 2. http://geodesic.mathdoc.fr/item/JIS_2009__12_2_a0/