In the general number field sieve, do we need to know whether powers of elements in the algebraic factor base divide an element $a+b\theta$? I'm reading this paper trying to implement the number field sieve.
http://citeseerx.ist.psu.edu/viewdoc/download?rep=rep1&type=pdf&doi=10.1.1.219.2389
Let $\theta$ be the root of some monic irreducible polynomial in $\mathbb{Z}[X]$ and let $A = \lbrace a_i + b_i \theta \rbrace $ be an algebraic factor base in $\mathbb{Z}[\theta]$. Let $l \in A$. For the purposes of the algorithm, when looking for $A$-smooth "elements" in the algebraic number ring $\mathbb{Z}[\theta]$ do we need to know if powers of $l$ divide a given element of $\mathbb{Z}[\theta]$? Or can we get away with only knowing if $l$ does?  
 A: You need to know the power of $l$ dividing the candidate $\alpha$ being considered for being smooth, but only modulo $2$.  That is, you only need to know if the power is odd or even.  If you collect enough of these smooth numbers, then by solving a certain system of equations modulo $2$, with coefficients determined by these powers, you can determine which smooth numbers to multiply to produce a perfect square.
Let me say that the paper you reference has some major flaws. (Edit: it appears to be a paper written by a student in the course ECE 575 at Oregon State U.)  There are numerous minor issues, like small omitted assumptions of nonzeroness, etc.  But as an algebraic number theorist, I can tell the paper is not written by someone with proper training in algebraic number theory.  The results the author is stating without proof are true only under very heavy assumptions -- that the set $\mathbb{Z}[\theta]$ defined on p.4 of your reference is the entire ring of integers in the algebraic number field $\mathbb{Q}(\theta)$, and that it forms a unique factorization domain.  Neither of these need hold with the setup he describes.
A much more carefully written paper is 
http://scholar.lib.vt.edu/theses/public/etd-32298-93111/materials/etd.pdf
The issues I just mentioned are summarized at the beginning of section 3.1.  He points out that early work on the SNFS did assume the conditions I mention above, and he gives the proper exposition of the results from algebraic number theory if you want to work without assuming these conditions, including some proof.
If you don't have the background for this paper, you could get it by reading some of the theory of commutative rings, up to prime and maximal ideals and properties of polynomial rings, and then reading a book on algebraic number theory, perhaps through ramification theory, but probably you could get away with less to understand the paper I linked.
