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156 CHAPTER 5. P -ADIC METHODS
written f (1) = g (1) h (1) (mod p 1 ) and our aim is to calculate a corresponding
factorization f (k) = g (k) h (k) (mod p k ) such that p k is sufficiently large.
Obviously, g (2) ≡ g (1) (mod p), and therefore we can write g (2) = g (1) +pĝ (2)
where ĝ (2) is a measure of the difference between g (1) and g (2) . The same holds
for f and h, so that f (2) = g (2) h (2) (mod p 2 ) becomes
Since f (1) = g (1) h (1)
f (1) + p ˆf (2) = (g (1) + pĝ (2) )(h (1) + pĥ(2) ) (mod p 2 ).
(mod p 1 ), this equation can be rewritten in the form
f (1) − g (1) h (1)
+
p
ˆf (2) = ĝ (2) h (1) + ĥ(2) g (1) (mod p). (5.5)
The left hand side of this equation is known, whereas the right hand side depends
linearly on the unknowns ĝ (2) and ĥ(2) . Applying the extended Euclidean
Algorithm (4) ) to g (1) and h (1) , which are relatively prime, we can find polynomials
ĝ (2) and ĥ(2) of degree less than g (1) and h (1) respectively, which satisfy
this equation modulo p. The restrictions on the degrees of ĝ (2) and ĥ(2) are
valid in the present case, for the leading coefficients of g (k) and h (k) have to be
1. Thus we can determine g (2) and h (2) .
Similarly, g (3) ≡ g (2) (mod p 2 ), and we can therefore write g (3) = g (2) +
p 2 ĝ (3) where ĝ (3) is a measure of the difference between g (2) and g (3) . The same
is true for f and h, so that f (3) = g (3) h (3) (mod p 3 ) becomes
Since f (2) = g (2) h (2)
f (2) + p 2 ˆf (3) = (g (2) + p 2 ĝ (3) )(h (2) + p 2 ĥ (3) ) (mod p 3 ).
f (2) − g (2) h (2)
Moreover, g (2) ≡ g (1)
(mod p 2 ), this equation can be rewritten in the form
p 2 + ˆf (3) = ĝ (3) h (2) + ĥ(3) g (2) (mod p). (5.6)
f (2) − g (2) h (2)
(mod p), so this equation simplifies to
p 2 + ˆf (3) = ĝ (3) h (1) + ĥ(3) g (1) (mod p).
The left hand side of this equation is known, whilst the right hand side depends
linearly on the unknowns ĝ (3) and ĥ(3) . Applying the extended Euclidean algorithm
to g (1) and h (1) , which are relatively prime, we can find the polynomials
ĝ (3) and ĥ(3) of degrees less than those of g (1) and h (1) respectively, which satisfy
this equation modulo p. Thus we determine g (3) and h (3) starting from g (2) and
h (2) , and we can continue these deductions in the same way for every power p k
of p until p k is sufficiently large.
We should note that Euclid’s algorithm is always applied to the same polynomials,
and therefore it suffices to perform it once. In fact, we can state the
algorithm in Figure 5.4.
To solve the more general problem of lifting a factorization of a non-monic
polynomial, we adopt the same solution as in the g.c.d. case: we impose the
leading coefficient in each factor to be the leading coefficient of the orgiinal
polynomial (which we may as well assume to be primitive). We also address the
problem of lifting n (rather than just 2) factors in Algorithm 26.