x8+ax+b=0

Simplest examples of irreducible and solvable by radicals octic trinomials, x⁸+ax+b, where a and b are integers. Checked range: |a|, |b| ≤ 23,600,000. Excluded trivial cases where a or b are equal zero.

Basic notes:

- if x⁸+ax+b=0 is solvable then x⁸-ax+b=0 is too. More generally x⁸+ax+b=0 is solvable if and only if x⁸+ac⁷x+bc⁸=0 is solvable (c is non-zero rational number).

- if x⁸+ax+b=0 is solvable then x⁸+(a/b)x⁷+(1/b)=0 is too. And in more general: x⁸+(ac/b)x⁷+(c⁸/b)=0

See incredible explicit formulas for the roots of the solvable polynomials.

See also the parameterization that I found.

Note 2: There is no other examples of group 8T44 (and 8T38 and few others too) for the range |a| ≤ 110,000,000, |b| ≤ 330,000,000.