In Rotman's Galois Theory, he defines separability for elements in field extensions as such:
Definition: If E/F is an extension, then $\alpha\in E$ is called separable if either it is transcendental or its irreducible polynomial is separable.
I have never heard of the irreducible polynomial for an element of a field? My assumption is that $\alpha$ is algebraic of degree $n$, then the irreducible polynomial that Rotman is referring to is the unique monic irreducible polynomial of degree $n$ $f(x) \in F[x]$ satisfying $f(\alpha) = 0$.
However, I'm unsure and couldn't find anything on Google. Could someone verify this for me?