The Stacks project

Lemma 43.23.7. 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 locus $C$ of non-invertible linear maps $V \to V$ is irreducible (note that since the determinant has degree $1$ in each variable, it is squarefree). 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. Details omitted. $\square$

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