What does it mean to find minimal polynomial of $\beta$ over some $F[\alpha]$?

Question

I am asked to find the minimal polynomial of $\beta$ over $F[\alpha]$. I understand what it means to find minimal polynomial of $\beta$ over $F[x]$ but what does it mean to find it over $F[\alpha]$ since this is not even a polynomial ring?

Answer 1

Given a field $K$, and an element $\beta$ that's algebraic over $K$, you can be tasked with finding the minimal polynomial of $\beta$ over $K$. If this $K$ happens to be an extension of a smaller field, such as $F[\alpha]$ which is an extension of $F$, that doesn't really change anything.

This minimal polynomial of $\beta$ will be an element of $K[x] = F[\alpha][x]$, but we do not say that it is the minimal polynomial "over" $K[x]$. It is the minimal polynomial of $\beta$ over $K = F[\alpha]$.

As a simple example, the minimal polynomial of $\sqrt[4]2$ over $\Bbb Q[\sqrt2]$ is $x^2 - \sqrt2$, which is an element of $\Bbb Q[\sqrt2][x]$. And the minimal polynomial of $\sqrt[3]5$ over $\Bbb Q[\sqrt[6]3]$ is $x^3 - 5\in \Bbb Q[\sqrt[6]3][x]$.