We consider the ideal $I=(x-y-1,x^2+y^2-2x)$ of $\mathbb{Q}[x,y]$ and $\mathbb{R}[x,y]$. We have to show if $I$ is a prime ideal in both ring. In $\mathbb{R}[x,y]$
I think is not a prime ideal, since: $$2x(x-y-1)-(x^2+y^2-2x)=x^2-y^2-2xy=x^2+y^2-2xy-2y^2=(x-y)^2-2y^2=(x-y-\sqrt{2}y)((x-y+\sqrt{2}y)) \in I$$ but neither $x-y-\sqrt{2}y$ or $x-y+\sqrt{2}y$ is in $I$, it is right? To prove that $I$ is a prime ideal in $\mathbb{Q}[x,y]$, I think to prove that $\mathbb{Q}[x,y]/I$ is a domain, but I don't see the right way.
In fact, $$\Bbb Q[x,y]/I\cong\Bbb Q[x]/(x^2+(x-1)^2-2x).$$ Then show that $2x^2-4x+1$ is irreducible over $\mathbb Q$ (respectively over $\mathbb R$).