  A \emph{chromatic root} is a root of the chromatic polynomial of a graph. Any
chromatic root is an algebraic integer. Much is known about the location of
chromatic roots in the real and complex numbers, but rather less about their
properties as algebraic numbers. This question was the subject of a seminar at
the Isaac Newton Institute in late 2008. The purpose of this paper is to report
on the seminar and subsequent developments.
  We conjecture that, for every algebraic integer $\alpha$, there is a natural
number $n$ such that $\alpha+n$ is a chromatic root. This is proved for
quadratic integers, an extension to cubic integers has been found by Adam Bohn.
The idea is to consider certain special classes of graphs for which the
chromatic polynomial is a product of linear factors and one "interesting"
factor of larger degree. We also report computational results on the Galois
groups of irreducible factors of the chromatic polynomial for some special
graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.
