Does every field extension of a field containing irreducible polynomial have a root?

Question

Suppose $F$ is a field and $K/F$ a field extension of $F$. Assume we have a polynomial $p(x) \in F[x]$ that is irreducible over $F$. My question is:

From these assumptions, does it follow that $p(x)$ has a root $c$ in $K$? If not, what would be a counter-example?

I have previously shown that $F[x] /( p(x) )$, where $( p(x) )$ denotes the principal ideal of $p(x)$, is a field extension of $F$ and contains a root of $p(x) $. Maybe this could be of help.

Thanks

Answer 1

A counter-example would be $x^2+1$, which is irreducible in $\Bbb Q[x]$ and which has no root in $\Bbb R$.