As title says, let $f(x,y) = x^2-y^2+1$ be a polynomial in 2 variables. Suppose $F = \mathbb{Z}/7\mathbb{Z}$, what can we say about the irreducibility of $f(x,y)$ in $F[x,y]$?
My thoughts are: $F[x,y]$ is UFD, therefore it mounts to check whether $f(x,y)$ is prime, which amounts to check whether $F[x,y]/\langle f(x,y)\rangle$ is an integral domain. How should I proceed and how to tackle such problems for other $F$? And other $f(x,y)$?
The easiest method to determine irreducibility of a multivariate polynomial is Eisenstein's criterion, which works over any field. To apply it here:
Note that $y - 1$ is a prime element in $F[y]$ (since $F[y]/(y-1) \cong F$), and $y-1 \mid y^2 - 1$, $(y-1)^2 \nmid y^2 - 1$. Thus by Eisenstein at $y-1$, $x^2 - (y^2-1)$ is irreducible in $F[y][x] = F[x,y]$.
As a bonus: $x^n + a(y^2 - 1)$ is irreducible in $F[x,y]$, for any $a \in F$, $n \in \mathbb{N}$. Also, since $y^2 + 1$ is irreducible (hence prime) in $F[y]$ (being a quadratic with no roots in $\mathbb{Z}/7\mathbb{Z}$), we also get irreducibility of $x^n + a(y^2 + 1)$ in $F[x,y]$.