If $f\in \mathbb{R}[X]$, then do only real numbers count as roots of $f$?

Question

I had an argument with a friend about this and I would like to know what you think. If we are given a polynomial $f\in \mathbb{R}[X]$, then do only real numbers count as roots of $f$?
Let me give you an example. Let's say we are given the polynomial $f=X^2+1 \in \mathbb{R}[X]$. It is obvious that $f(i)=f(-i)=0$, but are $i$ and $-i$ considered to be roots of $f$ since they are not in $\mathbb{R}$? I think that they are not and they would be only if we were told that $f\in \mathbb{C}[X]$.
EDIT: After the discussion in the comments, I came to the conclusion that a polynomial whose coefficients are in a field can also have roots in another field. Can you give me some examples of this occuring in polynomials which are not in $\mathbb{Z}[X],\mathbb{Q}[X],\mathbb{R}[X]$?

Answer 1

Unless you are talking to someone that has never been exposed to complex numbers, when you ask what are the roots of $p(x)\in\mathbb R[x]$, then most of the time (outside Real Analysis) it is implicit that you mean the complex roots. If you are interested only on real (or integer or whatever), you should state that explicitely.