Let $K$ be a field and $E$ an extension of $K$. Suppose that $a \in E$ is algebraic over $K$. Show that $K[a]$ is a field. (You can assume that $K[a]$ is an integral domain).
Since $a \in E$ is algebraic over $K$ there exists a polynomial $f \in K[x]$ such that$f(a)=0$. Now I think that if $f$ is irreducible, then $K[a] \cong K[x]/\langle f(x) \rangle$ so $K[a]$ is a field since $\langle f(x) \rangle$ is maximal?
This happens only if $f$ is irreducible which I don't know it is and $a \in E$ being algberaic doesn't imply that $f$ need to be irreducible.
Is there a missing condition here or can this be shown for $f$ not irreducible also?