The Stacks project

Lemma 37.26.4. Let $A$ be a domain with fraction field $K$. Let $P \in A[x_1, \ldots , x_ n]$. Denote $\overline{K}$ the algebraic closure of $K$. Assume $P$ is irreducible in $\overline{K}[x_1, \ldots , x_ n]$. Then there exists a $f \in A$ such that $P^\varphi \in \kappa [x_1, \ldots , x_ n]$ is irreducible for all homomorphisms $\varphi : A_ f \to \kappa $ into fields.

Proof. There exists an automorphism $\Psi $ of $A[x_1, \ldots , x_ n]$ over $A$ such that $\Psi (P) = ax_ n^ d +$ lower order terms in $x_ n$ with $a \not= 0$, see Algebra, Lemma 10.115.2. We may replace $P$ by $\Psi (P)$ and we may replace $A$ by $A_ a$. Thus we may assume that $P$ is monic in $x_ n$ of degree $d > 0$. For $i = 1, \ldots , n - 1$ let $d_ i$ be the degree of $P$ in $x_ i$. Note that this implies that $P^\varphi $ is monic of degree $d$ in $x_ n$ and has degree $\leq d_ i$ in $x_ i$ for every homomorphism $\varphi : A \to \kappa $ where $\kappa $ is a field. Thus if $P^\varphi $ is reducible, then we can write

\[ P^\varphi = Q_1 Q_2 \]

with $Q_1, Q_2$ monic of degree $e_1, e_2 \geq 0$ in $x_ n$ with $e_1 + e_2 = d$ and having degree $\leq d_ i$ in $x_ i$ for $i = 1, \ldots , n - 1$. In other words we can write

37.26.4.1
\begin{equation} \label{more-morphisms-equation-factors} Q_ j = x_ n^{e_ j} + \sum \nolimits _{0 \leq l < e_ j} \left( \sum \nolimits _{L \in \mathcal{L}} a_{j, l, L} x^ L \right) x_ n^ l \end{equation}

where the sum is over the set $\mathcal{L}$ of multi-indices $L$ of the form $L = (l_1, \ldots , l_{n - 1})$ with $0 \leq l_ i \leq d_ i$. For any $e_1, e_2 \geq 0$ with $e_1 + e_2 = d$ we consider the $A$-algebra

\[ B_{e_1, e_2} = A[\{ a_{1, l, L}\} _{0 \leq l < e_1, L \in \mathcal{L}}, \{ a_{2, l, L}\} _{0 \leq l < e_2, L \in \mathcal{L}}]/(\text{relations}) \]

where the $(\text{relations})$ is the ideal generated by the coefficients of the polynomial

\[ P - Q_1Q_2 \in A[\{ a_{1, l, L}\} _{0 \leq l < e_1, L \in \mathcal{L}}, \{ a_{2, l, L}\} _{0 \leq l < e_2, L \in \mathcal{L}}][x_1, \ldots , x_ n] \]

with $Q_1$ and $Q_2$ defined as in (37.26.4.1). OK, and the assumption that $P$ is irreducible over $\overline{K}$ implies that there does not exist any $A$-algebra homomorphism $B_{e_1, e_2} \to \overline{K}$. By the Hilbert Nullstellensatz, see Algebra, Theorem 10.34.1 this means that $B_{e_1, e_2} \otimes _ A K = 0$. As $B_{e_1, e_2}$ is a finitely generated $A$-algebra this signifies that we can find an $f_{e_1, e_2} \in A$ such that $(B_{e_1, e_2})_{f_{e_1, e_2}} = 0$. By construction this means that if $\varphi : A_{f_{e_1, e_2}} \to \kappa $ is a homomorphism to a field, then $P^\varphi $ does not have a factorization $P^\varphi = Q_1 Q_2$ with $Q_1$ of degree $e_1$ in $x_ n$ and $Q_2$ of degree $e_2$ in $x_ n$. Thus taking $f = \prod _{e1, e_2 \geq 0, e_1 + e_2 = d} f_{e_1, e_2}$ we win. $\square$


Comments (0)


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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0557. Beware of the difference between the letter 'O' and the digit '0'.