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Chapter 5
p-adic Methods
In this chapter, we wish to consider a different problem, that of factoring polynomials.
We will see that this cannot be solved by the methods of the previous
chapter, and we need a new technique for solving problems in (large) domains
via small ones. The basic idea behind these algorithms is shown in Figure 5.1:
instead of doing a calculation in some (large) domain R, we do it in several
smaller domains R i , pick one of these (say R l ) as the best one, grow the solution
to some larger domain ̂R l , regard this as being in R and check that it is
indeed the right result.
5.1 Introduction to the factorization problem
For simplicity, we will begin with the case of factoring a univariate polynomial
over Z. More precisely, we consider the following.
Problem 5 Given f ∈ Z[x], compute polynomials f i ∈ Z[x] (1 ≤ i ≤ k) such
that:
1. f = ∏ k
i=1 f i;
Figure 5.1: Diagrammatic illustration of Hensel Algorithms
R
k×reduce ↓
R 1
.
R k
calculation
- - - - - - - - - - - - - - - - - - - - - - - - - - - -> R
calculation
−→ R 1
⎫⎪ ⎬
choose
. .
calculation ⎪
−→ R ⎭
k
grow
−→ R l −→
↑ interpret
& check
̂R l
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