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Suppose we have the polynomial $x^2+1$.

If $p(x)\in \Bbb F[x]$ just means it has coefficients from $\Bbb F$

I read in a linear algebra book that if we consider $x^2+1 \in \Bbb R[x]$ then it has no roots, but if we consider $x^2+1 \in \Bbb C[x]$, then it has $i, -i$ as roots.

But what does the set of coefficients have to do with this? It is not the set of roots we mark with $\Bbb F[x].$ I'm a little confused as to what's going on here..

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    No, it's not the set of roots that "we mark with $\mathbb F[x]$". Saying that $p(x)$ is in $\mathbb F[x]$ means exactly that the coefficients of the monomials in $p(x)$ are in $\mathbb F$. – 2017-02-26
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    @Git Gud Instead, should we say simply that it has no real roots, but it has two complex roots? – 2017-02-26
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    @Michael Yes, "it has no roots" means "no roots in the coefficient field". – 2017-02-26

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