Let $p(x)$ be a minimal polynomial for $a$ for field $F$. This implies it is a monic polynomial of the lowest degree possible such that $p(x)=0$.
Why does $p(x)$ have to be irreducible? Why can't it be split into factors?
If $p(x)$ is reducible, which I think should be the case, then it can have some of its factors in common with other polynomials. Hence, $F[a]$ won't be a field then, in spite of $a$ being algebraic over $F$, because the polynomials which have non-zero factors in common with $p(x)$ will not have a multiplicative inverse.