  Let $R=k[x_1, ..., x_n]/(x_1^d + ... + x_n^d)$, where $k$ is a field of
characteristic $p$, $p$ does not divide $d$ and $n \geq 3$. We describe a
method for computing the test ideal for these diagonal hypersurface rings. This
method involves using a characterization of test ideals in Gorenstein rings as
well as developing a way to compute tight closures of certain ideals despite
the lack of a general algorithm. In addition, we compute examples of test
ideals in diagonal hypersurface rings of small characteristic (relative to $d$)
including several that are not integrally closed. These examples provide a
negative answer to Smith's (2000, Comm. in Alg.) question of whether the test
id eal in general is always integrally closed.

Commutative Algebra