There is a paper "Factoring integers with the number field sieve" (download it here, for example).

I can't understand how they reason the correctness of computing ideal valuations in the case of using non-monic non-linear polynomials, which is given on pages 50-51 of the PDF above.

Here are the excerpts I don't understand:

Assume that $f(X,Y) = c_d X^d + c_{d-1} X^{d-1} Y^1 + ... + c_{1} X Y^{d-1} + c_{0} Y^d$.

Actually this one is pretty good. I was able to prove that $A$ is closed under multiplication and that $A=\mathbb{Z}[\alpha]\cap\mathbb{Z}[\alpha^{-1}]$.

The real problems emerge when it comes to the next excerpt:

I have the following questions:

How do first degree prime ideals of $A$ look like (what are their generators)? And how do I deduce it from that mappings of $\alpha$ and $\alpha^{-1}$?

How do I test if a number belongs to a first degree prime ideal of $A$?

- Why do we use $A + \alpha A$ instead of just $A$? How $A + \alpha A$ is related to $A$ and its first degree prime ideals?

So, technically, I don't understand the last excerpt at all.

I would be very grateful if someone explains it to me or gives a good literature reference.