Answer by Rene Schipperus for Is $Th(\mathbb{Z}[x])$ uncountably categorical?

Since you contain the arithmetic of the integers you can pick some infinite family of primes and say that an element of degree $0$ is divisible by those primes and not any other primes. This gives uncountably many types over a countable set, so the theory is not $\aleph_0$ stable and thus not uncountably categorical.