When does a polynomial in $GF$ have a multiplicative inverse?

Question

When does a polynomial in $GF$ have a multiplicative inverse?

Are there values of $n$ such that all polynomials in $GF(n)$ have multiplicative inverses?

EDIT: To address the comments, I mean:

1. All coefficients are in $GF(n)$
2. Addition and multiplication of polynomials is defined pointwise: $(f+g)(x) := f(x) + g(x)$ and $(f \cdot g)(x) := f(x) \cdot g(x)$.

I believe that definition solves the ambiguity raised. If it's not the standard terminology, please help me learn the right terms to use!

Answer 1

A non-constant polynomial can never have an inverse in the ring of polynomials over any field. To see this, note that if $f$ and $g$ are non-zero polynomials of degrees $n$ and $m$ then $fg$ is a polynomial of degree $n + m$.