$x^m+y^m+1$ is irreducible in $k[x,y]$

Question

Question:

Let $k$ be a field with characteristic $0$. Let $m\geq 2$ be an integer. Show that $f(x,y)=x^m+y^m+1$ is irreducible in $k[x,y]$.

Answer:

I have no idea how to solve this question. Any hint/help would be appreciated. Thanks in advance...

Answer 1

You can use Eisenstein criterion. Indeed, the polynomial $X^m + 1$ is separable because $\partial(X^m + 1) = mX^{m - 1}$ share no root with $X^m + 1$. Therefore, it has no square factor so if $p$ is any prime polynomial with coefficients in $k$ that divides $X^m + 1$, then $p^2$ doesn't divide it. We deduce that $Y^m + X^m + 1$ is a $p$-Einsenstein polynomial in $(k[X])[Y] = k[X,Y]$ thus it is irreducible.