  We introduce a new algebra associated with a hyperplane arrangement
$\mathcal{A}$, called the Solomon-Terao algebra $\mbox{ST}(\mathcal{A},\eta)$,
where $\eta$ is a homogeneous polynomial. It is shown by Solomon and Terao that
$\mbox{ST}(\mathcal{A},\eta)$ is Artinian when $\eta$ is generic. This algebra
can be considered as a generalization of coinvariant algebras in the setting of
hyperplane arrangements. The class of Solomon-Terao algebras contains
cohomology rings of regular nilpotent Hessenberg varieties. We show that
$\mbox{ST}(\mathcal{A},\eta)$ is a complete intersection if and only if
$\mathcal{A}$ is free. We also give a factorization formula of the Hilbert
polynomials when $\mathcal{A}$ is free, and pose several related questions,
problems and conjectures.
