Polynomials are some of the most general objects in mathematics and polynomial rings form some of the most useful structures in algebra. Often, when faced with one, being able to reduce it to its most basic components, its irreducible factors, makes things much easier and tells us a lot about the polynomial itself and the concepts it relates to, in cases as varied as the characteristic polynomial of a matrix, a polynomial ring quotient and so on.

Unfortunately, our usual method of factoring polynomials tends to be quite slow (especially with regard to running it on a computer) and, as such, it's important to find better and quicker ways of doing it. A particularly interesting example is that of polynomials with coefficients in . To simplify them, we can multiply through the coefficients to obtain an equivalent polynomial with coefficients in and reducing the coefficients modulo a prime gives a polynomial over the finite field . Factoring polynomials over finite fields is many times more efficient than doing it over or and what's interesting is that the original factors over can be recovered from the factorisation over a finite field, making this a very attractive approach. Join us this Thursday at 7 pm in MS.04 as Richard Howell-Peak takes us through some ways of doing this and explains how the method can be used to factor polynomials into irreducibles (as always, no prior knowledge of any of these concepts is assumed), after which we head on to our regular pub social.