Irreducibility of the Tutte Polynomial of a Connected Matroid

We solve in the affirmative a conjecture of Brylawski, namely that the Tutte polynomial of a connected matroid is irreducible over the integers. If M is a matroid over a set E, then its Tutte polynomial is defined as T(M; x, y)= C A ı E (x − 1) r(E) − r(A) (y − 1) | A | − r(A) , where r(A) is the rank of A in M. This polynomial is an important invariant as it contains much information on the matroid; see [2, 3] for useful surveys. One of the basic properties of T(M; x, y) is that, if M is the direct sum of two matroids M 1 and M 2 , then T(M; x, y)=T(M 1 ; x, y) T(M 2 ; x, y). In particular, this implies that T(M; x, y) has a non-trivial factor in Z[x, y] if M is disconnected. Brylawski [1] conjectured that the converse also holds; this paper is devoted to a proof of this conjecture.