I was solving the following exercise:
Find the inverse of $p(x) = 1 + x$ in $R[x]$ over $\mathbb Z/5\mathbb Z$ or show that it does not exist.
and finding that it does not exist because if there is a polynomial $q(x)$ such that $p(x)q(x)=1$. Which will contradict the proposition:
Let $R$ be a commutative ring. For every pair of non-zero polynomials $p$ and $q$ in $R[x]$, if the leading coefficient of $p$ or $q$ is not a zero divisor in $R$ then $\deg(pq)=\deg(p)+\deg(q)$
my question is it safe to assume that there does not exist an inverse of polynomial of $\deg>0$ over $\mathbb Z/p\mathbb Z$ where $p$ is a prime number?