Higher degree case

There is no such general formula for the roots of polynomials of degree $5$ or greater. This was proven by an astonishing display of ingenuity and creativity by E. Galois before he died at the age of 23. We will not prove this fact here, however, see §5.16.1 below. For the curious, the polynomial $x^5-x+1$ has roots which cannot be expressed in terms of radicals of rational numbers, whereas $x^5-5x+12$ has roots which can so be expressed.