Wilson's theorem states that a natural number $n > 1$ is a prime number if and only if the product of all the positive integers less than $n$ is one less than a multiple of $n$.

Is there a version of the theorem in $\mathbb Z/p\mathbb Z[t]$ where $p$ is a prime?

Are there any other non-trivial versions?

My motivation is the following.

There is no space or time efficient algorithm for primes without Cramer's conjecture but perhaps working with irreducible polynomials requires no unproven conjectures and so is finding an irreducible in $\mathbb Z/p\mathbb Z[t]$ any easier?

Wilson's theorem states that a natural number $n > 1$ is a prime number if and only if the product of all the positive integers less than $n$ is one less than a multiple of $n$.

Is there a version of the theorem in $\mathbb Z/p\mathbb Z[t]$ where $p$ is a prime?

Are there any other non-trivial versions?

My motivation is the following.

There is no efficient algorithm for primes without Cramer's conjecture and so perhaps working with irreducible polynomials avoids conjectures and so is finding irreducibles in $\mathbb Z/p\mathbb Z[t]$ any easier?

Wilson's theorem states that a natural number $n > 1$ is a prime number if and only if the product of all the positive integers less than $n$ is one less than a multiple of $n$.

Is there a version of the theorem in $\mathbb Z/p\mathbb Z[t]$ where $p$ is a prime?

Are there any other non-trivial versions?

Wilson's theorem states that a natural number $n > 1$ is a prime number if and only if the product of all the positive integers less than $n$ is one less than a multiple of $n$.

Is there a version of the theorem in $\mathbb Z/p\mathbb Z[t]$ where $p$ is a prime?

Are there any other non-trivial versions?

My motivation is the following.

There is no space or time efficient algorithm for primes without Cramer's conjecture but perhaps working with irreducible polynomials requires no unproven conjectures and so is finding an irreducible in $\mathbb Z/p\mathbb Z[t]$ any easier?