Why Is Inverse of a Polynomial Still a Polynomial? The fourth paragraph of the wikipedia article algebraic extension states that $K[a]$ is a field, which means every polynomial has an inverse. The inverse has to be a polynomial over $K$ as well. It seems it requires $K$ to be a non-integral domain. How do we resolve the confusion? How do we prove this paragraph? 

Polynomials over a field $K$, that is, elements of $K[x]$, don't generally (unless they have degree zero) have an inverse.
$K[a]$ is a different object, it is isomorphic to the quotient ring $K[x]/(m)$ where $m$ is the minimal polynomial of $a$ (you can see this via the first isomorphism theorem using the morphism $\varphi:K[x]\to K[a]$ which sends $\sum k_i x^i\mapsto\sum k_i a^i$), assuming $a$ lives inside a field extension of $K$ its minimal polynomial will be irreducible, so the ideal $(m)$ it generates will be maximal and the quotient $K[x]/(m)$ will be a field.
Inverses in this field can be computed via the usual euclidean algorithm, take a polynomial $f\in K[x]$ with $\gcd(f,m)=1$, then by Bezout's theorem there are $h$ and $g$ such that $fh+gm=1$, but in the quotient ring $gm=0$ so $h$ is the inverse of $f$
