Final steps in the General Number Field Sieve I try to understand the General Number Field Sieve, based on Michal Case paper "A beginner's guide to the general number field sieve". I was able to reproduce some results from the example, i.e. $$\prod_{(a,b)\in V} (a+bm) = 459997127517955195582606376960000$$ and $$\prod_{(a,b)\in V}(a+b\theta)=58251363820606365*\theta^{2}+149816899035790332*\theta+75158930297695972$$
Now I need to compute square roots in $\mathbb{Z}$ and $\mathbb{Z}[\theta]$. But the next step result from the paper seems to be incorrect, since $$459997127517955195582606376960000 \neq 2553045317222400^{2}$$
How should I compute both square roots to get the results presented in the paper? How to calculate square root in $\mathbb{Z}[\theta]$? Next, how can I find the mapping $\phi$ required in the next step of the algorithm?
 A: Based on answer to another question (In the general number field sieve, do we need to know whether powers of elements in the algebraic factor base divide an element $a+b\theta$?) we know that the mentioned paper has some major and minor errors. 
Matthew E. Briggs in his thesis An Introduction to the General Number Field Sieve describes the method to successfully perform the computations. It sums up to the following algorithm:


*

*Determining applicable finite fields



The first stage in computing $x = φ(β) \pmod{p}$ (...) is to find a number of finite fields that are “compatible” with $\mathbb{Q}(\theta)$, which boils down to finding prime integers $p$ for which $f(x)$ remains irreducible modulo $p$.



*Finding square roots in finite field

*Using the CRT - once square roots in the finite fields are known, the Chineses Remainder Theorem can be used to complete the computations.


Notice that this approach eliminates the need to compute an explicit value of square root in $\mathbb{Z}[\theta]$. As stated by the author, in practice we never calculate this value.
The second value is calculated simply as square root in $\mathbb{Z}$.
