Polynomials

Linear and quadratic polynomial equations are easy to solve. Since 16th century there are also known expressions for the roots of cubic and quartic equations. But for a long time there were unknown similar methods for polynomials of higher degrees. At the turn of 18th and 19th centuries Paulo Ruffini, Niels Abel and Évariste Galois proved that it is impossible to solve by radicals polynomial equations of degree 5 or higher except of special cases when the roots of given polynomial have some particular symmetry.

Here are presented simplest examples of irreducible and solvable polynomials of degrees greater than 4. All results were got using GAP, especially the Radiroot package. Excluded trivial examples such as x⁵+a=0 or x⁶+ax⁴+bx²+c=0.

There is single subpage with, I hope, interesting results for non-irreducible solvable polynomials.

Summary of sieving for irreducible but solvable trinomials can be found here.