Lemma 43.23.6. Let $k$ be a field. Let $n \geq 1$ be an integer and let $x_{ij}, 1 \leq i, j \leq n$ be variables. Then

is an irreducible element of the polynomial ring $k[x_{ij}]$.

Lemma 43.23.6. Let $k$ be a field. Let $n \geq 1$ be an integer and let $x_{ij}, 1 \leq i, j \leq n$ be variables. Then

\[ \det \left( \begin{matrix} x_{11}
& x_{12}
& \ldots
& x_{1n}
\\ x_{21}
& \ldots
& \ldots
& \ldots
\\ \ldots
& \ldots
& \ldots
& \ldots
\\ x_{n1}
& \ldots
& \ldots
& x_{nn}
\end{matrix} \right) \]

is an irreducible element of the polynomial ring $k[x_{ij}]$.

**Proof.**
Let $V$ be an $n$ dimensional vector space. Translating into geometry the lemma signifies that the variety $C$ of non-invertible linear maps $V \to V$ is irreducible. Let $W$ be a vector space of dimension $n - 1$. By elementary linear algebra, the morphism

\[ \mathop{\mathrm{Hom}}\nolimits (W, V) \times \mathop{\mathrm{Hom}}\nolimits (V, W) \longrightarrow \mathop{\mathrm{Hom}}\nolimits (V, V),\quad (\psi , \varphi ) \longmapsto \psi \circ \varphi \]

has image $C$. Since the source is irreducible, so is the image. $\square$

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