Question about the proof of the existence of a splitting field

Question

In the proof given here https://en.wikipedia.org/wiki/Splitting_field, how are we talking about the roots of, for example, $p(x)$ in $F[x]$ quotient the ideal generated by $p(x)$? Isn't $p(x) = 0$ in this field? Then how do we discuss its roots? I thought if a polynomial were the zero polynomial in some field $G$ then it doesn't make sense to speak about its roots in $G$.

Answer 1

This is about names of variables. Consider, for instance, the complex numbers, conventionally $\Bbb R[i]/(i^2+1)$, where we say that the real polynomial $x^2+1$ splits.

In short, the variable name you use to expand the field is now taken, so you can't use it as the generic variable name in a polynomial. So if you have a polynomial $p(x)$, which is irreducible over a field $F$ (which is to say, it is irreducible in the polynomial ring $F[x]$), then over the field $F[x]/(p(x))$, the polynomial $p(t)$ has at least one root (which is to say, it is reducible in the polynomial ring $(F[x]/(p(x)))[t]$).

(Note that $F[x]/(p(x))$ is not necessarily a splitting field for $p$. You are guaranteed the existence of one root, but nothing more. In some cases you do get all roots, but not in general.)