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Lemma 9.11.2. Two polynomials in $k[x]$ are relatively prime precisely when they have no common roots in an algebraic closure $\overline{k}$ of $k$.

Proof. The claim is that any two polynomials $P, Q$ generate $(1)$ in $k[x]$ if and only if they generate $(1)$ in $\overline{k}[x]$. This is a piece of linear algebra: a system of linear equations with coefficients in $k$ has a solution if and only if it has a solution in any extension of $k$. Consequently, we can reduce to the case of an algebraically closed field, in which case the result is clear from what we have already proved. $\square$

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