I have been trying to learn some Galois Theory from Weintraub's book of the same name. Weintraub uses the word “root” of a polynomial without ever defining exactly what it is.
I understand that a root of $f\in k[X]$ is some element $\alpha$ st $f(\alpha)=0$. But my problem is, to which field/ring/set is $\alpha$ the element of?
I think it should be defined as “Let $f\in k[X]$. A root of $f$ is any $\alpha$ in any field extension $L$ of $k$ st $f(\alpha)=0$.“
Is this right? If it is, is it also correct to define the root of $f\in A[X_1,....,X_n]$, $A$ any ring, as any $\alpha:=(\alpha_1,....,\alpha_n)$ in any ring extension of $A$ st $f(\alpha)=0$?