A problem regarding a maximal ideal in a polynomial ring in several variables

Question

$\mathbf {The \ Problem \ is}:$ Is the ideal $I =\langle x^2-2,y^2+1,z\rangle$ maximal in the polynomial ring $R =\mathbb Q[x,y,z]$ ?

$\mathbf {My \ approach} :$ Actually, by $3rd$ isomorphism theorem of rings, quotenting both $R$ and $I$ by $\langle z \rangle$, we get $\frac{R}{I} \cong \frac{\mathbb Q[x,y]}{\langle x^2-2 , y^2+1\rangle}$ ;

Now , if we define a map $\phi : \mathbb Q[x,y] \to \mathbb Q(\sqrt 2,i)$ where $i^2=-1;$ by $\phi(f(x,y)) = f(\sqrt 2,i)$ then can we show that kernel of this map is $J =\langle x^2-2,y^2+1\rangle$ ?

Here, one inclusion is obvious, but how about the other ?

I have tried a lot, but I can't prove it . A small hint is warmly appreciated.

Answer 1

Hint. Let $f\in\mathbb{Q}[x,y]=\mathbb{Q}[y][x]=\mathbb{Q}[x][y]$. Using long division by $x^2+1$, we get $f=q(x,y)(x^2+1)+r_1(y)+r_2(y)x$. Now, divide $q$ and $r_1,r_2$ by $y^2-2$.