One of the most famous applications of representation theory is Burnside's Theorem, which states that if p and q are prime numbers and a and b are positive integers, then no group of order paqb is simple. In the first edition of his book Theory of groups of finite order (1897), Burnside presented group-theoretic arguments which proved the theorem for many special choices of the integers a, b, but it was only after studying Frobenius's new theory of group representations that he was able to prove the theorem in general. Indeed many later attempts to find a proof which does not use representation theory were unsuccessful, until H. Bender found one in 1972.

A preliminary lemma

We prepare for the proof of Burnside's Theorem with a lemma (31.2) which is concerned with character values. In order to establish this lemma we require some basic facts about algebraic integers and algebraic numbers, which we now describe. We omit proofs of these – for a good account, see for instance the book by Pollard and Diamond listed in the Bibliography.

An algebraic number is a complex number which is a root of some non-zero polynomial over ℚ. We call a polynomial in x monic if the coefficient of the highest power of x in it is 1.

Let α be an algebraic number; and let p(x) be a monic polynomial over ℚ of smallest possible degree having α as a root.