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Let $K[X_{1,1},..., X_{i,j}...X_{n,n}]$ a polynomial ring in $n^2$ variables $X_{i,j}$. The determinant in $X_i$ is the expansion ("the Leibniz formula")

$$\det(X_{1,1},..., X_{i,j},...,X_{n,n})=\sum _{\sigma \in S_{n}}\left(\operatorname {sgn}(\sigma )\prod _{i=1}^{n}X_{i,\sigma _{i}}\right)$$

Is there a standard approach to prove that the determinant regarded as a homogeneous polynomial in variables $X_{i,j}$ is irreducible?

I found a couple questions here (Determinant of symmetric matrix is an irreducible polynomial and Slick proof the determinant is an irreducible polynomial) asking essentially about the same issue but I'm wondering if there exist a classical 'model' approach to prove this claim, since all answers I found there seem not to be standard.

user267839
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  • I don't know of a "classical 'model' approach", but my intuition is that linear homogeneous polynomials in many variables are almost never reducible since there are so few ways they could possibly factor. In that sense I'd speculate that any classical approach would be similar to the top answers to your linked questions--check to see if there's any hope of a factorization and rule it out combinatorially by cases. – Joshua P. Swanson Jan 29 '21 at 00:23

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