  The Tutte polynomial is originally a bivariate polynomial enumerating the
colorings of a graph and of its dual graph. But it reveals more of the internal
structure of the graph like its number of forests, of spanning subgraphs, and
of acyclic orientations. In 2007, Ardila extended the notion of Tutte
polynomial to the hyperplane arrangements, and computed the Tutte polynomials
of the classical root systems for a certain prime power of the first variable.
In this article, we compute Tutte polynomials of ideal arrangements. Those
arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined
for ideals of root systems. For the ideals of the classical root systems, we
bring a slight improvement of the finite field method showing that it can
applied on any finite field whose cardinality is not a minor of the matrix
associated to a hyperplane arrangement. Computing the minor set associated to
an ideal of a classical root system permits us particularly to deduce the Tutte
polynomials of the classical root systems. For the ideals of the exceptional
root systems of type $G_2$, $F_4$, and $E_6$, we use the formula of Crapo.
