I have a non zero polynomial $f\in F[X]$ where $F$ is a field. Let $L$ be a field extension of $F$ so that $f$ splits completely in $L[X]$, so $f(X)=c\prod_{i=1}^n (X-a_i)$ with $c,a_i\in L$.
If the roots $a_i$ of $f$ are all distinct then why is the same property true for any other field $L'$ over which $f$ splits completely?
I've tried looking for a proof in algebra books but I haven't been able to find one.
Any help (proof or a reference) will be much appreciated.
Let $F$ be a field, and $f\in F[X]$ a polynomial. The presence/absence of multiple roots for $f$ in any extension field can be decided without ever needing to exit the field $F$. This is because a univariate polynomial has multiple zeros, if and only if $\gcd(f,f')$ is non-trivial. This can be found in many algebra texts. A logical place is when separability is discussed, but it may come earlier. That greatest common divisor can be computed by Euclid's algorithm, and a glance at the algorithm tells us that the operations never produce coefficients $\notin F$. Therefore the algorithm runs the same way in any extension field of $F$.
Consequently if $f$ and $f'$ have no common zeros in some extension field of $F$, they won't have common zeros in any other extension field of $F$ either.