# Showing an irreducible polynomial is a minimal polynomial.

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.

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What's your definition of minimal polynomial? –  Git Gud Jan 13 '13 at 15:32

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.