abstract algebra - Intuition about turning a polynomial ring into a field

Let's consider the polynomial ring $R=F[x]$ over a field $F$. Then by taking the quotient by a principal ideal $I=(f(x))$ generated by an irreducible polynomial $f(x)$, we obtain a field $R'=F[x]/(f(x))$.

It's easy to see that $R'$ is indeed a field. Since the ideals of $R$ which contain $I$ are in bijective correspondence with the ideals of $R'$, we can conclude that $R'$ has only two ideals and is therefore a field (as $I$ is maximal in $R$ since $f(x)$ is irreducible).

I wanted to ask, is there an intuitive way of understanding why taking the quotient by some ideal makes $F[x]$ into a field? I would ideally like some way of demonstrating that the existence of a nonzero polynomial equivalent to zero in $R'$ somehow allows us to describe an algorithm to calculate multiplicative inverses...