TITLE: How do we show that an ideal of polynomials is prime
QUESTION [6 upvotes]: I'm trying to solve this exercise:

To do so, I'm trying to prove that $(X_1^2+X_2^2+X_3^2)$ is a prime ideal. 
Suppose now $f,g\in \mathbb R[X_1,X_2,X_3]$ and $f\cdot g\in (X_1^2+X_2^2+X_3^2)$, i. e., $f\cdot g=h\cdot(X_1^2+X_2^2+X_3^2)$, with $h\in \mathbb R[X_1,X_2,X_3]$.
Suppose $f\notin (X_1^2+X_2^2+X_3^2)$, then we have to prove that $g\in (X_1^2+X_2^2+X_3^2)$...
I couldn't go further, I really need help.
Thanks a lot

REPLY [2 votes]: Alternative: It suffices to prove that $\mathbb{R}[x,y,z]/(x^2+y^2+z^2) \otimes_{\mathbb{R}} \mathbb{C}$ is an integral domain. But this is isomorphic to
$\mathbb{C}[x,y,z]/(x^2+y^2+z^2) \stackrel{y \mapsto iy}{\cong} \mathbb{C}[x,y,z]/((x+y)(x-y)+z^2) \cong \mathbb{C}[u,v,z]/(uv+z^2)$
which is obviously an integral domain (or use Eisenstein with the prime element $u$).