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5.5. HENSEL LIFTING 155
polynomials are, nevertheless, “quite rare”, but [ABD85] showed that they can
occur in the manipulation of algebraic numbers.
Even if we have factorizations modulo several primes, a further problem
arises, which we will illustrate with the example of x 4 + 3. This factors as
and
x 4 + 3 = ( x 2 + 2 ) (x + 4) (x + 3) mod 7
x 4 + 3 = ( x 2 + x + 6 ) ( x 2 + 10 x + 6 ) mod 11. (5.1)
In view of the second factorization, the first has too much decomposition, and
we need only consider the split
x 4 + 3 = ( x 2 + 2 ) ( x 2 + 5 ) mod 7, (5.2)
obtained by combining the two linear factors.
When we come to combine these by Chinese Remainder Theorem (Theorem
35) to deduce a congruence modulo 77, we have a dilemma: do we pair
(
x 2 + x + 6 ) with ( x 2 + 2 ) or ( x 2 + 5 ) ? Both seem feasible.
In fact, both are feasible. The first pairing gives
and the second gives
x 4 + 3 = ( x 2 + 56 x + 72 ) ( x 2 − 56 x − 16 ) mod 77, (5.3)
x 4 + 3 = ( x 2 + 56 x + 61 ) ( x 2 − 56 x − 5 ) mod 77 : (5.4)
both of which are correct. The difficulty in this case, as in general, is that,
while polynomials over Z 7 have unique factorization, as do those over Z 11 (and
indeed modulo any prime), polynomials over Z 77 (or any product of primes) do
not, as (5.3) and (5.4) demonstrate.
5.5 Hensel Lifting
Our attempts to use the Chinese Remainder Theorem seem doomed: we need a
different solution, which is provided by what mathematicians call p-adic methods,
and computer algebraists call Hensel Lifting. This is the topic of the next
three sections.
5.5.1 Linear Hensel Lifting
This is the simplest implementation of the phase described as ‘grow’ in Figure
5.1: we grow incrementally Z p → Z p 2 → Z p 3 → · · · → Z p m.
For simplicity, we consider first the case of a monic polynomial f, which
factorizes modulo p as f = gh, where g and h are relatively prime (which implies
that f modulo p is square-free, that is, that p does not divide the resultant of f
and f ′ ). We use parethesized superscripts, as in g (1) , to indicate the power of
p modulo which an object has been calculated. Thus our factorization can be