$Gcd(f,g)\neq 1$ implies that $f$ and $g$ have a common root in some extension of the base field

Question

I have to prove that for two polynomials $\;f,g \in F[x]\;$ , we have that $\; gcd(f,g) \neq 1\;$ iff there exists $\;a \in K\;$ which is in some extension field of $F$ and $f(a)=g(a)=0$

I tried to use Euclid's algorithm and to get the gcd of the polynomials but what is its properties?

Answer 1

If $\gcd(f,g)=1$, then there exist polynomials $u,v$ with $uf+vg=1$, hence $f$ and $g$ cannot have a root in common.

If $\gcd(f,g)=h$ is a non-constant polynomial, then any root of $h$ (in a suitable extension field $K$) is of course also a root of $f$ and of $g$.