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150 CHAPTER 5. P -ADIC METHODS
2. each f i is irreducible in Z[x], i.e. any polynomial g that divides f i is either
an integer or has the same degree as f i .
We might wonder whether we wouldn’t be better off considering f i ∈ Q[x], but
in fact the answers are the same.
Proposition 48 Any factorization over Q[x] of a polynomial f ∈ Z[x] is (up
to rational multiples) a factorization over Z[x].
Proof. Let f = ∏ k
i=1 f i with f i ∈ Q[x]. By clearing denoninators and removing
contents, ( we can write f i = c i g i with g i ∈ Z[x] and primitive and c i ∈ Q. Hence
∏k
) (
f =
i=1 c ∏k
i i=1 i)
g , and, since the product of primitive polynomials is
primitive (Lemma 2), ∏ k
i=1 c i is an integer, and can be absorbed into, say, g 1 .
Even knowing that we have only to consider integer coefficients does not
seem to help much — we still seem to have an infinite number of possibilities
to consider. In fact this is not quite the case.
Notation 20 Let the polynomial f = ∑ n
i=0 a ix i to be factored have degree n,
and coefficients bounded by H. Let us suppose we are looking for factors of
degree at most d.
Corollary 12 (to Theorem 27) It is sufficient to look for factors of degree
d ≤ n/2, whose coefficients are bounded by 2 d H.
One might have hoped that it was sufficient to look for factors whose coefficients
are bounded by H, but this is not the case. [Abb09] gives the example of
f = x 80 − 2x 78 + x 76 + 2x 74 + 2x 70 + x 68 + 2x 66 + x 64 + x 62 + 2x 60 + 2x 58
−2x 54 + 2x 52 + 2x 50 + 2x 48 − x 44 − x 36 + 2x 32 + 2x 30 + 2x 28 − 2x 26
+2x 22 + 2x 20 + x 18 + x 16 + 2x 14 + x 12 + 2x 10 + 2x 6 + x 4 − 2x 2 + 1
whose factors have coefficients as large as 36, i.e. 18 times as large as the
coefficients of f. Non-squarefree polynomials are even worse: [Abb09, p. 18]
gives the example of
−1 − x + x 2 − x 3 + x 4 + x 5 + x 6 + x 8 + x 10 − x 11
−x 12 − x 13 + x 14 − x 15 − x 17 − x 18 + x 20 + x 21 =
(
1 + 4 x + 8 x 2 + 14 x 3 + 21 x 4 + 28 x 5 + 34 x 6 + 39 x 7 + 42 x 8 + 43 x 9 + 41 x 10
+37 x 11 + 32 x 12 + 27 x 13 + 21 x 14 + 15 x 15 + 9 x 16 + 4 x 17 + x 18) (x − 1) 3
Although some caution is in order, his table appears to show coefficient growth
behaving like 0.7 × 1.22 d , where d is the degree of a polynomial with coefficients
at most ±1.
Corollary 13 To detect all irreducible factors of f (except possibly for the last
one, which is f divided by all the factors of degree ≤ n/2), it suffices to consider
(
2 d H ) d+1
polynomials.
We can in fact do better, since the leading coefficient of the factor must divide
a n , and similarly the trailing coefficient must divide a 0 , so we get 2 d(d−1) H d+1 ,
and in practice such “brute force” methods 1 can easily factor low-degree poly-
1 Sometimes known as Newton’s algorithm.