TITLE: Irreducibility criteria for polynomials with several variables.
QUESTION [5 upvotes]: Let $K$ be a field. Show that $x^2-yz$ is irreducible in $K[x,y,z]$. Deduce that $x^2-yz$ is prime.

If it is $K[x]$, then there are several methods which can be used to check whether a given polynomial is irreducible. But how do we check that when we have a polynomial of several variables? No idea how to do it. Any theorems? Moreover, in general, it is not true that irreducible elements are prime. So, how can I deduce the last result? Any help is appreciated.

REPLY [5 votes]: If it is reducible, you can write it as the product of two linear polynomials. For degree reasons it has to be $p(x,y)q(x,z)$. Now
$$x^2-yz=(ax+by)(cx+dz)\iff ac=1,\ bd=-1,\ ad=0,\ bc=0.$$
These equations are incompatible.
In UFDs, irreducible elements are prime.