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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)