GCD of Polynomials does not depend on Base Field

Theorem
Let $F\subset E$ be fields.

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

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

Then $R=S$.

In particular, $S\in F[X]$.

Proof
By definition of greatest common divisor, $R\mid S$ in $E[X]$.

By Polynomial Forms over Field is Euclidean Domain, there exist $A,B\in F[X]$ with $AP+BQ=R$.

Because $S\mid A,B$, $S\mid R$ in $E[X]$.

By $R\mid S$ and $S\mid R$, $R=S$.