I've been trying to show that the sphere $x^2+y^2+z^2=1$ is irreducible as an algebraic set over $\mathbb{R}$. By this I mean that it cannot be written as the union of two proper algebraic subsets.
It is not enough that the polynomial is irreducible (as shown in this answer for example), and we can't use tools such as the Nullstellensatz since $\mathbb{R}$ is not algebraically closed.
Maybe we can use the fact that it is a manifold, or that it is connected. I might be missing something obvious, but I haven't been able to make any progress.
Any help would be much appreciated.
It’s smooth and connected. This is sufficient over any field.