Write the determinant as a polynomial expression in the elementary symmetric polynomials

Question

How to write $\det\begin{bmatrix}x_1&x_2&x_3&x_4\\x_2&x_3&x_4&x_1\\x_3&x_4&x_1&x_2\\x_4&x_1&x_2&x_3 \end{bmatrix}$in terms of elementary symmetric polynomials ?

if $R:=\mathbb F_2$ and $n=4$

The determinant is $-x_1^4+x_2^4-x_3^4+x_4^4+4(x_1^2x_2x_4-x_1x_2^2x_3-x_1x_3x_4^2+x_2x_3^2x_4)+2(x_1^2x_3^2-x_2^2x_4^2)$

So if we have $\mathbb F_2$ can I ignore the last two terms, then:

$(x_2^4-x_1^4)+(x_4^4-x_3^4)=(x_2^2+x_1^2)(x_2+x_1)(x_2-x_1)+(x_4^2+x_3^2)(x_4+x_3)(x_4-x_3)$

How can I continue ?

Answer 1

Over $\mathbb F_2$, your first expression (assuming it is correct, I did not check) reduces to $(x_1 + x_2 + x_3 + x_4)^4$.

Hope that helps,