We just got into algebraic extensions, and this one threw me for a loop.
Left $f$ be a polynomial irreducible over $F$, and let $E$ be an extension field of $F$ in which $f$ has root $\alpha$. Show that $f$ is a minimal polynomial for $\alpha$ over $F$.
I don't even know where to start so I can't tell you what I've done.
Best Answer
Hint: a minimal polynomial of $\alpha$ divides in $F[X]$ every polynomial vanishing at $\alpha$. Indeed, such polynomials form an ideal and $F[X]$ is principal.