Confused about Finite fields and polynomials

Question

I'm asked to give a polynomial that has a root over a finite field but not a root over R. My understanding is that the finite field is contained in R (more restrictive) so how can there be a root in one but not the other?

Answer 1

Finite fields have a characteristic which is not 0, which creates other things. For example, consider $f(x)=x^2+1$. Clearly in $\mathbb R$, this has no roots. But if you are in the field of 2 elements, then 1 is a root, as $1+1=0$

Answer 2

If you mean $\mathbb R$, then no finite fields aren't contained in $\mathbb R$. To see this, note that finite fields have characteristic $p$ for some prime $p$, i.e. $p=0$. However this is certainly not true over the real numbers, which have characteristic $0$.

Like Alan said, $x^2+1$ has no roots over $\mathbb R$, but does have roots over the field with $p$ elements whenever $p \equiv 1 \pmod{4}$.