The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

9.11 Relatively prime polynomials

Let $K$ be an algebraically closed field. Then the ring $K[x]$ has a very simple ideal structure as we saw in Lemma 9.10.2. In particular, every polynomial $P \in K[x]$ can be written as

\[ P = c(x - \alpha _1) \ldots (x - \alpha _ n), \]

where $c$ is the constant term and the $\alpha _1, \ldots , \alpha _ n \in k$ are the roots of $P$ (counted with multiplicity). Clearly, the only irreducible polynomials in $K[x]$ are the linear polynomials $c(x - \alpha )$, $c, \alpha \in K$ (and $c \neq 0$).

Definition 9.11.1. If $k$ is any field, we say that two polynomials in $k[x]$ are relatively prime if they generate the unit ideal in $k[x]$.

Continuing the discussion above, if $K$ is an algebraically closed field, two polynomials in $K[x]$ are relatively prime if and only if they have no common roots. This follows because the maximal ideals of $K[x]$ are of the form $(x - \alpha )$, $\alpha \in K$. So if $F, G \in K[x]$ have no common root, then $(F, G)$ cannot be contained in any $(x - \alpha )$ (as then they would have a common root at $\alpha $).

If $k$ is not algebraically closed, then this still gives information about when two polynomials in $k[x]$ generate the unit ideal.

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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