How to show elements in $F[x]/p(x)$ are closed under division?

Question

I know that $K = F[x]/p(x)$ is a field when $p(x)$ is irreducible, but how do I show from the addition, and multiplication mod $p(x)$ structure that every element in this field extension are actually closed under division? namely, when I take an element $r(x)$ in $K$, how do I know that division by another element in K yields another element in $K$? Because in general F[x] is not closed under division, while F(x) is. And I know that $K \cong F(\alpha)$ where the latter is a field, but exactly, when considering the element of a field extension in their polynomial representation, how can they be closed under division?

Answer 1

How explicit do you want this to be?

In any PID, let's say $R$, there is a well-defined notion of GCD and it holds that if $a,b\in R$ then the ideal $(a,b)$ is equal to the ideal $(d)$ where $d=\gcd(a,b)$. Now $F[x]$ is a PID and if we take any nonzero element $\overline{f(x)}\in F[x]/(p(x))$ then this is the residue of some $f(x)\in F[x]$ which is not divisible by $p(x)$. Since $p(x)$ is irreducible it follows that $\gcd(f,p)=1$ and hence $(f,p)=F[x]$ so in particular there exist polynomials $g(x)$ and $h(x)$ such that $f(x)g(x)+p(x)h(x)=1$. Going back modulo $p(x)$, we get that $\overline{f(x)}\cdot\overline{g(x)}=\overline1$, so $\overline{f(x)}$ has an inverse in $F[x]/(p(x))$.