I'm re-reading the primitive element lemma and I can't reason the following concept. Let $f,g\in F[x]$ be in the polynomial ring of one variable over the field $F$. Let those two polynomials have a unique common simple root $\beta$. Then the largest unary common divisor of $f$ and $g$, with coefficients in $F$ is $x-\beta$.
How come $x-\beta\in F[x]$?
Euclid's algorithm for calculating the gcd of $f$ and $g$ will never leave the ring $F[x]$. Because $F$ is a field you can normalize the result to be monic (=leading coefficient equal to one) in the end.