I want to show that the polynomial $P(x,y,z) = z^3+3x^2y+3xy^2+3x^2z+3xz^2+3y^2z+3yz^2+6xyz$ is irreducible. I know that I can consider this as a polynomial in one variable z with coefficients in $\mathbb{k}[x, y]$, and I can apply Eisensteins with this in mind, but is it suitable to use $p=3$ here? If $\mathbb{k}= \mathbb{Q}$, is it not possible that given appropiate values of $x$ or $y$, Eisensteins would fail at $p=3$? If I cannot use $p=3$, what are my options?
2026-03-25 15:59:56.1774454396
Correct application of eisenstein's criterion?
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