If we have a polynomial in a polynomial ring, are its "roots" only valid if the roots are in the ring itself?
i.e. For $x^2+1 \in \mathbb{C}[x]$ we have roots $i$ and $-i$. But if we consider it as $x^2+1 \in \mathbb{R}[x]$, because $i, -i \notin \mathbb{R}$, do we say that $x^2+1$ has no roots?
On
I wouldn't. I'd just say it has no roots in $\mathbb R$. It's very often useful to consider roots of a member of a polynomial ring that lie in an extension of the coefficient ring.
It's almost always worthwhile to increase clarity at the expense of spilling a few drops of ink. Your readers will thank you.
This is just a matter of context. Sometimes "roots" refers to roots only in the base ring, and other times it refers to roots in extensions of the base rings (typically, in some fixed but possibly unspecified algebraic closure). If you have any doubt, it's good to disambiguate by saying things like "roots in $\mathbb{R}$" or "roots in an algebraic closure". You will encounter authors who don't always disambiguate though and in most cases it is clear from context; for instance, if someone refers to a polynomial as having no roots, it is usually obvious that they must mean in the base ring.