Let $F$ be a field. Let $p_1$ and $p_2$ be two separable polynomial in $F[x]$, then is it true that $\gcd(p_1,p_2) \in F[x]$?
2026-04-04 09:40:14.1775295614
Is $\gcd(p_1,p_2) \in F[x]$
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It is always true that $\gcd(p_1,p_2)$ is in $F[x]$, because Euclid's algorithm for calculating a $\gcd$ works essentially unchanged using polynomial arithmetic. Separability doesn't enter into it.