This is actually a theorem from my lecture notes and it says:
Suppose $L/K$ is field extension, and $f,g$ $\in$ $K[X]$. Then gcd($f,g$) = $1$ in $K[X]$ if and only if gcd($f,g$) = $1$ in $L[X]$
The proof in the lecture notes is just some lines on how the euclidean algorithm, which computes the gcd doesn't take into account the coefficients of the polynomials, but I would like to see if there is a mathematical proof for this statement.
I don't quite see how the statement would work, for e.g., suppose f and g have only one common factor ($x$-$\alpha$), where $\alpha$ $\in$ L, then surely the hcf of f, g in K[X] is 1 and in L[X] is ($x$-$\alpha$).
(Or is it me that has got the concept of hcf wrong? The above statement obviously assumes that for two polynomials in K[X] to have a factor, the common factor must also be in K[x].?)
The Euclidean Algorithm argument that you refer to is quite mathematical. Alternately, you can use the fact that if $f$ and $g$ are relatively prime polynomials in $K[X]$, then there are polynomials $u$ and $v$ in $K[X]$ such that $fu+gv=1$ (the Bezout "Identity"). This forces $f$ and $g$ to be relatively prime when considered as polynomials in $L[X]$.
Your "$x-\alpha$" argument assumes that $\alpha$ is the only common root of $f$ and $g$ in $L$.