I have found a few resources such as 2.0.3 here. There however are a series of "identifications" such as saying $k[x,y,z] = k[y,z][x]$
Why can I make these identifications? can I ALWAYS do so? I see that in the polynomial in their example $x^2 + y^2 + z^2$ that we have the same degree polynomial and there is no term with mixed variables. What if I wanted to check reducibility on something such as $xy + z^2$?
The example goes on to say that it suffices to show that $y^2 + z^2$ is divisible by some prime $p$ in $k(z)[y]$, and not by $p^2$. Thus It suffices to show that $y^2 + z^2$ is not a unit, and has no repeated factor, in $k(z)[y]$. So it seems as if we are just taking the polynomials in $k(z)$ and appending the variable y to it, is this correct i.e. $k(z) + a \cdot y$?
Now I understand Eisenstein's criteria but all of the resources I find only apply it in a single variable - otherwise they mention that it "suffices to show" and I fail to see the connection. I could really use a walkthrough of how these work, more concretely, a walkthrough on why $xy + z^2$ is irrreducible.