Show $X_1^5+X_1^2X_2+X_1X_2+X_2$ is irreducible

Question

For a field $K$, is the polynomial $X_1^5+X_1^2X_2+X_1X_2+X_2$ irreducible in $K[X_1,X_2]$?

I think I have to show it for $K(X_1)[X_2]$ but I don't know how to do that.

Answer 1

• Use Eisenstein's criterion with the ideal $\langle X_2\rangle.$
• Now you can say that your polynomial is irreducible in $K(X_1)[X_2]$.
• Use that the polynomial is primitive to conclude that it is irreducible in $K[X_1X_2]$

Answer 2

Hint: Use Eisenstein's criterion (from left to right $X_2\nmid1$, $X_2\mid X_2$, $X_2\mid X_2$ and $X_2^2\nmid X_2$). Then use Gauss' lemma to pass back from the fraction field (the polynomial is primitive).