finite fields - Minimum polynom of an element in $K=mathbb{F}_5[x]/(x^2-2)$

I want to know how to calculate the minimum polynom of an element $\alpha$ in $K=\mathbb{F}_5[X]/(X^2-2)$ where $\alpha$ is the image on $K$ of $X+2$

I'm already verficated that $K$ is a field. As I know, the minimum polynom is $g$ with the smallest degree satisfying $g(\alpha)=0$.

In my notes there is an indication: If $f(X)=X^2-2$, calculate $f(X-2)$.

But I don't understand at all. I would very appreciate any help. Thanks.

Answer

Let's write $\beta$ for the residue class of $X$ in ${\mathbb F}_5[X]/(X^2-2)$, so we're looking at the field ${\mathbb F}_5[\beta]$. What is the minimum polynomial of $\beta$ over ${\mathbb F}_5$? (Trivial!)

Now $\alpha = \beta + 2$. So how do you now turn the minimum polynomial of $\beta$ into the minimum polynomial of $\alpha$? (This is what the hint tells you to do.)