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Let $A_n$ be the $n \times n$-Vandermonde matrix (see for example https://en.wikipedia.org/wiki/Vandermonde_matrix )viewed as a matrix over the fraction field of the polynomial ring over a field $K$ (we can assume characteristic 0 if that helps) in the variables $x_1,...,x_n$.

Question 1: Is it true that the characteristic polynomial of $A$ is irreducible (at least when the field is $\mathbb{Q}$)?

This seems to be true for $n \leq 5$.

Question 2: Are all the coefficients of the characteristic polynomial irreducible in $K[x_1,...,x_n]$ except the last one corresponding to the determinant?

Question 3: If Question 1 has a positive answer, what is the Galois group of that polynomial? Is there a nice expression of the roots in an algebraic closure ?

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    $\begingroup$ If we set $x_i=i$, then the characteristic polynomial of the Vandermonde matrix is irreducible over $\mathbb{Q}$ for $1\leq i\leq 100$. I also checked that the Galois group is $S_n$ for $1\leq n\leq 9$. $\endgroup$
    – Richard Stanley
    Commented Jul 11, 2021 at 23:25
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    $\begingroup$ Oops, I meant $1\leq n\leq 100$. $\endgroup$
    – Richard Stanley
    Commented Jul 12, 2021 at 0:11

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