  In this paper, we prove the existence of a first-order definition of the
polynomial ring over a nonprincipal ultraproduct of finite fields of unbounded
cardinalities in its fraction field by a universal-existential formula in the
language of rings augmented by an additional constant symbol $t$. As a
consequence, we prove that the full first-order theory of the rational function
field over a nonprincipal ultraproduct of finite fields of characteristic $0$
is undecidable.
