Proving that the inverse of an algebraic element is algebraic Let $E$ be an extension of a field $F$. Suppose $c \in E$ is algebraic over $F$ where $c \neq 0$. I want to prove that $c^{-1}$ is also algebraic over $F$. 
I feel like I'm missing something obvious here and that it shouldn't be very difficult to do. I know that $c^{-1} \in F(c)$, but that's about it. Do I actually need to construct a polynomial over $F$ with root $c^{-1}$ (presumably from the minimum polynomial of $c$)? Any hints would be appreciated.

Hints: for $\,c\neq 0\,$ (of course)
$$0=f(c)=\sum_{k=0}^na_kc^k=c^n\sum_{k=0}^n\frac{a_k}{c^n}c^k\ldots$$
