I'm trying to prove this ideal:
$$(x^2+y^2+z^2+x+y+z,\ x^5+y^5+z^5+2(x+y+z),\ x^7+y^7+z^7+3(x+y+z))\subset \mathbb C[x,y,z]$$
can't be maximal.
In order to do so, I'm using the Nullstellensatz theorem and showing this ideal is not of this form: $(x-a_1,y-a_2,z-a_3)$, where $a_i\in \mathbb C$.
The problem I don't how to do this.