Galois theory

Galois theory gives a beautiful insight into the classical problem of when a given polynomial equation in one variable, such as x^5-3x^2 4=0 has solutions which can be expressed using radicals. Historically the problem of solving algebraic equations is one of the great drivers of algebra, with the quadratic equation going back to antiquity, and the discovery of the cubic solution by Italian mathematicians in the 1500’s. Here we look at the quartic equation and give a method for factoring it, which relies on