$x^a+ y^b + z^c$ is irreducible in $\mathbb C[x,y,z]$

Question

Let $a,b,c$ be positive integers. Then $f = x^a + y^b + z^c$ is irreducible in $\mathbb{C}[x,y,z]$.

By Gauss, $f$ is irreducible in $\mathbb{C}[x,y,z]$ iff is so in $\mathbb{C}(z)[x,y]$, and so iff in $\mathbb{C}(y, z)[x]$.

So by Eisenstein, it is sufficient to show that $y^b + z^c$ has a single prime factor.

If $b=c$, this factors through $\Pi(y + \zeta ^i z)$ for some primitive root of unity $\zeta$, so ok.

But if $b\neq c$?

This question is related to this post.

Thank you very much!

Answer 1

Let $f\in\Bbb{C}[y,z]$ be a factor of $y^b+z^c$ with multiplicity $m\geq1$, and let $g\in\Bbb{C}[y,z]$ be such that $y^b+z^c=f^mg$. Then taking derivatives with respect to $y$ and $z$ shows that $$by^{b-1}=mf^{m-1}f_yg+g_yf^m=f^{m-1}(mgf_y+g_yf),$$ $$cz^{c-1}=mf^{m-1}f_zg+g_zf^m=f^{m-1}(mgf_z+g_zf).$$ In particular we see that $f^{m-1}$ divides both $by^{b-1}$ and $cz^{c-1}$, and hence it is a constant. This shows that $y^b+z^c$ has no repeated irreducible factors. Hence by Eisensteins criterion, the polynomial $$x^a+y^b+z^c\in\Bbb{C}[y,z][x],$$ is irreducible as it is Eisenstein w.r.t. every irreducible factor of $y^b+z^c$.