### V.21 The Insolubility of the Quintic

*Martin W. Liebeck*

Every student will be familiar with the formula for the roots of a quadratic polynomial *ax*^{2} + *bx* + *c*, namely (-b ± )/2*a*. Perhaps less familiar is the fact that there is also a formula for the roots of a cubic: write the cubic as *x*^{3} + *ax*^{2}+ *bx*+*c*, and make the substitution *y* = *x* + *a* to rewrite it in the form *y*^{3} + 3*hy* + *k*. The roots of this are then of the form

While the quadratic formula was known ...