what is meant "Express the following as polynomials in α of degree at most 2"?

Question

I've been studying Ring and field theory online recently and I came upon a confusing question today.

Let α be a root of the polynomial $X^3 + 4X + 2$ over $Q$. What is the degree of $Q(α)/Q$? Express the following as polynomials in α of degree at most 2.

(a) $α^4$

(b) $(α + 1)^3$

(c) $(α + 1)^{-1}$

I understand the first part. the degree of the extension is obviously 3 but I don't know what is meant by "in alpha" isn't this just a root ? secondly what do they mean by expressing alphas as a polynomial ? maybe it's just lack of sleep and this is more obvious than it seems right now but any help would be appreciated. I think if i could even see one example I'd be able to get a handle on it .

Answer 1

For (a), you can find polynomials $q(x), r(x)$ s.t. $x^{4} = (x^{3}+4x+2)p(x)+r(x)$ with $\deg r\leq 2$. If we put $x=\alpha$, we get $\alpha^{4}=r(\alpha)$, so $r(x)$ is the desired polynomial. You can do the same way for (b).

For (c), you can find $x^{3}+4x+2 = (x+1)s(x)+c$ for some $s(x)\in \mathbb{Q}[x]$ and $c\in \mathbb{Q}$, then for $x=\alpha$, $0=(\alpha+1)q(\alpha)+c$.