  We describe a connection between the subjects of cluster algebras, polynomial
identity algebras and discriminants. For this, we define the notion of root of
unity quantum cluster algebras and prove that they are polynomial identity
algebras. Inside each such algebra we construct a (large) canonical central
subalgebra, which can be viewed as a far reaching generalization of the central
subalgebras of big quantum groups constructed by De Concini, Kac and Procesi
and used in representation theory. Each such central subalgebra is proved to be
isomorphic to the underlying classical cluster algebra of geometric type. When
the root of unity quantum cluster algebra is free over its central subalgebra,
we prove that the discriminant of the pair is a product of powers of the frozen
variables times an integer. An extension of this result is also proved for the
discriminants of all subalgebras generated by the cluster variables of nerves
in the exchange graph. These results can be used for the effective computation
of discriminants. As an application we prove an explicit formula for the
discriminant of the integral form over ${\mathbb{Z}}[\varepsilon]$ of each
quantum unipotent cells of De Concini, Kac and Procesi for arbitrary
symmetrizable Kac-Moody algebras, where $\varepsilon$ is a root of unity.
