In the second edition of Artin's algebra book, page 472, the following exercise is given:
Let $\alpha$ be a complex root of $x^3-3x+4$. Find the inverse of $\alpha^2+\alpha+1$ in the form $a\alpha^2+b\alpha+c$, with $a$, $b$, $c$ in $\mathbb{Q}$.
The exercise itself is not difficult. One can use the extended Euclidean Algorithm or brute force to find $a$, $b$ and $c$. I do not understand how the fact that $\alpha$ is a complex root of $x^3-3x+4$ relevant. As far as I can see, both the approaches do not make use of the fact that $\alpha$ is a complex root. There is possibly a simple explanation, but it eludes me. Can someone explain why this condition is there? Is it possible that the above condition gives a shorter way of solving the exercise?
Well, if you just said "a root of $x^3-3x+4$", then it would be unclear what field this root is supposed to be living in. So "complex" specifies that we are looking at a root in $\mathbb{C}$, as opposed to (say) a root in $\mathbb{F}_{37}$. As it turns out, it doesn't really matter that the field is specifically $\mathbb{C}$, but it does matter that it is some field of characteristic $0$ as opposed to a field of positive characteristic (and requiring $a,b,c\in\mathbb{Q}$ would not make sense in the latter case anyways).