Let $F$ be a field. Let $p_1$ and $p_2$ be two separable polynomial in $F[x]$, then is it true that $\gcd(p_1,p_2) \in F[x]$?
Is $\gcd(p_1,p_2) \in F[x]$
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abstract-algebra
polynomials
finite-fields
extension-field
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3Double-check the question, it's analogous to asking if squarefree integers have a gcd (of course they do, since *all* integers have a gcd). Maybe you want to know if the gcd remains separable? – 2017-02-16
1 Answers
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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.