In other words, how do I check whether the algebraic set given by a set of polynomial equations is irreducible? For example, I encountered the following example when working through Vakil's notes on algebraic geometry:
Show that $k[x,y,z,w]/(wz-xy,wy-x^2,xz-y^2)$ is an integral surface. (Here $k$ is a field.)
I know that if the ideal is principal, then we only have to check that the generator is prime. However, I don't know of any efficient way to check primeness in general. How does one approach such problems? In particular, how do I show that $(wz-xy,wy-x^2,xz-y^2)$ is a prime ideal? Thanks in advance.