Geodesic
On a sequence of nonsolvable quintic polynomials
Journal of integer sequences, Tome 12 (2009) no. 2
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 
@article{JIS_2009__12_2_a0,
     author = {Johnstone,  Jennifer A. and Spearman,  Blair K.},
     title = {On a sequence of nonsolvable quintic polynomials},
     journal = {Journal of integer sequences},
     year = {2009},
     volume = {12},
     number = {2},
     zbl = {1223.12005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2009__12_2_a0/}
}
TY  - JOUR
AU  - Johnstone,  Jennifer A.
AU  - Spearman,  Blair K.
TI  - On a sequence of nonsolvable quintic polynomials
JO  - Journal of integer sequences
PY  - 2009
VL  - 12
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/JIS_2009__12_2_a0/
LA  - en
ID  - JIS_2009__12_2_a0
ER  - 
%0 Journal Article
%A Johnstone,  Jennifer A.
%A Spearman,  Blair K.
%T On a sequence of nonsolvable quintic polynomials
%J Journal of integer sequences
%D 2009
%V 12
%N 2
%U http://geodesic.mathdoc.fr/item/JIS_2009__12_2_a0/
%G en
%F JIS_2009__12_2_a0
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/