Galois Theory and the Quintic Equation

Most students know the quadratic formula for the solution of the general quadratic polynomial in terms of its coefficients. There are also similar formulas for solutions of the general cubic and quartic polynomials. In these three cases, the roots can be expressed in terms of the coefficients using only basic algebra and radicals. We then say that the general quadratic, cubic, and quartic polynomials are solvable by radicals. The question then becomes: Is the general quintic polynomial solvable by radicals? Abel was the first to prove that it is not. In turn, Galois provided a general method of determining when a polynomial’s roots can be expressed in terms of its coefficients using only basic algebra and radicals. To do so, Galois studied the permutations of the roots of a polynomial. We will use the result that the Galois group of a polynomial is solvable if the polynomial is solvable by radicals to show that the general quintic is not solvable by radicals.