$x^2 +y^2 + z^2$ is irreducible in $\mathbb C [x,y,z]$ 
Is $x^2 +y^2 + z^2$ irreducible in $\mathbb C [x,y,z]$?

As $(x^2+y^2+z^2)= (x+y+z)^2- 2(xy+yz+zx)$,
$$(x^2+y^2+z^2)=\left(x+y+z+\sqrt{2(xy+yz+zx)}\right)\left(x+y+z-\sqrt{2(xy+yz+zx)}\right).$$
But how to show that none of these factors belong to $\mathbb C [x,y,z]$?

We can consider $X^2+Y^2+Z^2$ as a polynomial in $R[X]$, where $R=\mathbb C[Y,Z]$. Now apply the Eisenstein criterion with the irreducible $Y+\mathrm iZ$ in $R$:

*

*It divides $Y^2+Z^2$;

*It does not divide $1$, the coefficient of $X^2$;

*It does not divide $(Y^2+Z^2)^2$.

Irreducibility follows.
