Polynomials are one of the most fundamental algebraic objects in mathematics, yet they often appear to have many surprising properties - an example is a polynomial in many variables whose set of positive values is precisely the set of primes. Galois theory restricts its attention to polynomials of one variable, which are much better behaved.

As such, Galois theory first arose through the study of polynomial equations of small degree - after the familiar quadratic formula (known to the Babylonians), Tartaglia and Cardano were the first to find the cubic formula for the roots of a cubic polynomial in 1535, and Cardano's student, Ferrari, found the quartic equation in 1545. But a quintic equation remained elusive despite many mathematicians' attempts at trying to find it - indeed, in 1799 Ruffini sketched a proof of the non-existence of such a formula, which was fleshed out in 1824 by Abel, and such a discovery was one of the main motivations for Galois theory.
To prove this theorem, Ruffini extensively studied permutations of roots of polynomials, which initiated the study of Galois theory a few decades later when Galois considered groups as a generalisation of permutations, hence allowing statements about polynomials and fields to be translated into algebraic statements.

Join us at 7:30 in MS.04 as Sam Derbyshire explains why the quintic isn't soluble by radicals, why the 65 537-gon is constructible by ruler and compass and more. After which we construct our way to the pub.