GCD of Polynomials does not depend on Base Field

Theorem
Let $F\subset E\subset L$ be fields.

Let $P,Q,R\in F[X]$ be polynomials.

Let $\gcd(P,Q)=R$ in $E[X]$.

Then:
 * $\gcd(P,Q)=R$ in $F[X]$
 * $\gcd(P,Q)=R$ in $L[X]$