$x^2 +y^2 + z^2$ 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 $\mathbb C [x,y,z]$?

Let, $i$ be a square root of $-1$ in $K$.Then we have, $x^2+y^2+z^2=(x+iy)(x-iy)+z^2$
Since $x+iy$ is irreducible in $K[x,y]$ and $x+iy$ does not divide $x-iy;$ irreducibly follows from the Eienstein's Citeria in $K[x,y,z]=K[x,y][z]$
