Irreducibility in multiple dimensions

[Simfish]

So we know some of the irreducibility criteria when we have one dimension X.

But what about multidimensional abstract algebra?

From Dummit Foote, we get that we can pair up every power of X with the powers of Y that happen to be associated with that power of X, and then treat those powers of Y as coefficients of X. Do we then use the same steps that we use in our standard irreducibility criterion? (other than the division by the ideal (xy) - which can lead to degenerate cases?)

 

[morphism]

There are analogues of Gauss's lemma and Eisenstein's criterion for any polynomial ring R[x] over a UFD R. (In particular, for R[x,y]=(R[x])[y].) I'm not sure if this is what you're asking though.