8 Hensel's Lemma and Applications
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7.8 Hensel’s Lemma and Applications
399
Theorem 7.8.1 (Hensel’s Lemma) Let f (x) = c0 + c1 x + · · · + cn x n be a polynomial in Z p [x] (coefficients are p-adic integers). Let f (x) be the formal derivative of
f (x). Suppose a 0 ∈ Z p with f (a 0 ) ≡ 0 mod p and f (a 0 ) ≡ 0 mod p. Then, there
exists a unique p-adic integer a such that f (a) = 0 and a ≡ a 0 mod p.
As preparation for the proof of Hensel’s lemma we recall Newton’s method for
solving a non-linear equation f (x) = 0 over the reals where f (x) is a differentiable
real-valued function. We start with an initial guess x0 . This initial guess must be
sufficiently close to a solution for this method to work but we will ignore this here
and refer to [A] for the technical requirements. Given x0 we form the tangent line to
the curve y = f (x) at the point (x0 , f (x0 )). This has the equation
y − f (x0 ) = f (x0 )(x − x0 ).
Let x1 be where the tangent line crosses the x-axis, that is where y = 0. We then
have
f (x0 )
− f (x0 ) = f (x0 )(x1 − x0 ) =⇒ x1 = x0 −
f (x0 )
assuming that f (x0 ) = 0. This provides the initial step in an iteration scheme. Consider the tangent line at (x1 , f (x1 )) and obtain
x2 = x1 −
and in general
xn+1 = xn −
f (x1 )
assuming f (x1 ) = 0
f (x1 )
f (xn )
assuming f (xn ) = 0.
f (xn )
Under appropriate conditions (see [A]) this iteration scheme will converge to a solution of f (x) = 0. How close the initial guess must be to a solution for the method
to converge depends on the function f (x) (see [A]).
This method can be applied to polynomial equations P(x) = 0 over the reals. The
proof of Hensel’s lemma in the p-adic field Q p utilizes a p-adic version of Newton’s
technique.
Proof Let f (x) be an p-adic integral polynomial, that is, f (x) has p-adic coefficients,
and let a 0 be as in the statement of Hensel’s lemma. We will prove the existence of
a solution a by inductively constructing its canonical p-adic expansion
a = d0 + d1 p + · · · + dk p k + · · ·
where di are p-adic digits to be determined. Let ak be the k-th convergent for a,
ak = d0 + d1 + · · · + dk p k .
7 The Fields Q p of p-Adic Numbers: Hensel’s Lemma
400
We will use an induction and a p-adic version of Newton’s method to show that
we can find p-adic digits so that f (ak ) ≡ 0 mod p k+1 and ak ≡ a 0 mod p. Then as
ak → a we have a as the desired solution.
Let a 0 have the canonical p-adic expansion
a 0 = b0 + b1 p + · · · + bk p k + · · ·
Take a0 = d0 = b0 . Then a0 ≡ a 0 mod p and f (a0 ) ≡ 0 mod p. This establishes the
lowest level of an induction.
Now, suppose we have ak−1 satisfying f (ak−1 ) ≡ 0 mod p k and ak−1 ≡ a 0 mod
p. Now let
ak = ak−1 + dk p k
where dk is a p-adic digit to be determined. Then
n
f (ak ) = f (ak−1 + dk p k ) =
ci (ak−1 + dk p k )i .
i=0
Then
n
f (ak ) = c0 +
i
ci (ak−1
+ i(aki−1
dk p k + terms in powers higher than p k+1 )).
1
i=1
This implies that
f (ak ) = f (ak−1 ) + dk p k f (ak−1 ).
By the inductive hypothesis we have f (ak−1 ) ≡ 0 mod p k and hence there is a p-adic
digit ek with
f (ak ) = ek p k + dk p k f (ak−1 ).
To obtain the appropriate digit dk we must then have
ek + dk f (ak−1 ) ≡ 0 mod p.
Since ak−1 ≡ a 0 mod p we have f (ak−1 ) ≡ f (a 0 ) ≡ 0 mod p. Therefore, the digit
dk can be found by
ek
mod p
dk = −
f (ak−1 )
and hence f (ak ) ≡ 0 mod p. Notice that approximating the p-adic digits uses essentially the same iteration scheme as Newton’s method over the reals.
7.8 Hensel’s Lemma and Applications
401
Now consider
a = d0 + d1 p + · · · + dk p k + · · ·
Since f (a) ≡ f (ak ) mod p k+1 for all k we must have f (a) = 0.
Now assume that ak−1 has the desired properties and consider ak . Let dk be a
p-adic digit to be determined and consider
ak = ak−1 + dk p k .
The uniqueness of a follows from the uniqueness of the sequence of convergents ak .
The proof of Hensel’s lemma provides an algorithm for constructing the solution
to an equation f (x) = 0 with f (x) ∈ Z p [x]. This algorithm is analogous to Newton’s
Method for solving real polynomial equations.
Suppose a 0 is a solution to f (x) ≡ 0 mod p. Then follow the procedure outlined
in the proof. Take d0 the first p-adic digit of a 0 and let a0 = a 0 . Let ak = ak−1 + dk p k
k−1
and iteratively find the digits dk by dk = f−a
for k ≥ 1.
(ak−1 )
Theorem 7.8.2 A polynomial with rational integer coefficients (in Z[x]) has a root
in Z p if and only if it has an integer root modulo p k for any k ≥ 1.
Proof Suppose that f (x) ∈ Z[x] and suppose that f (a) = 0 where a ∈ Z p . Then
from the proof of Hensel’s lemma there exists a sequence of integers (ak ) with ak ≡ a
mod p k . Since f (ak ) ≡ f (a) mod p k and f (a) = 0 we must have an integer solution
mod p k for each k.
Conversely, suppose that for each k there is an integer ak with f (ak ) ≡ 0 mod
p k . We have seen that the p-adic integers are complete so the sequence ak has a
convergent subsequence (ak ). Suppose that the limit of this subsequence is a. A
polynomial is a continuous function on any normed field (see exercises) and hence
f (a) = lim f (ak ).
However f (ak ) ≡ 0 mod p k for all k and therefore f (a) ≡ 0 mod p k for all k and
hence f (a) = 0.
Corollary 7.8.1 If a polynomial F(x) with integer coefficients has no roots modulo
p then it has no roots.
Hensel’s lemma can be used to describe the roots of unity in Q p .
Theorem 7.8.3 For any prime p and (m, p) = 1 there exists a primitive m-th root
of unity in Q p if and only if m|( p − 1). In this case every m-th root of unity is also a
( p − 1)-th root of unity. The set of ( p − 1)-th roots of unity forms a cyclic subgroup
of U (Z p ) of order p − 1.
402
7 The Fields Q p of p-Adic Numbers: Hensel’s Lemma
Proof If m|( p − 1) then p − 1 = km and hence every m-th root of unity in Q p is
also a ( p − 1)-th root of unity. Consider the polynomial f (x) = x p−1 − 1. Then its
formal derivative is f (x) = ( p − 1)x p−2 . Now let a be a rational integer with 1 ≤
a ≤ p − 1. Then from Fermat’s theorem, we have f (a) = 0 and further f (a) = 0
since | f (a)| p = 1. Therefore Hensel’s lemma implies that there are exactly p − 1
solutions to f (x) = 0 and they are all ( p − 1)-th roots of unity.
Conversely suppose that a ∈ Q p with a m = 1 then |a| p = 1 and a is a p-adic
integer. Let · · · a1 a0 . = a then a ≡ a0 mod p and hence a0m = 1. Since a0 is a rational
integer this implies that m|( p − 1).
The set of ( p − 1)-th roots of unity in Q p is then a finite subgroup of a field and
as we saw in Theorem 2.4.13 this must be cyclic.
As we have seen in this book, quadratic residues modulo a prime are important in
several different areas of number theory. In fact determining quadratic residues was
crucial in the Rabin encryption system. The final result of this section ties quadratic
residues modulo a prime p to square roots in the p-adic integers.
Lemma 7.8.1 A rational integer a not divisible by p has a square root in Z p ( p = 2)
if and only if a is a quadratic residue modulo p.
Proof Let a ∈ Z with (a, p) = 1. Consider the polynomial P(x) = x 2 − a in Z p [x].
Suppose that a is a quadratic residue mod p. Then there exists a 0 with a 0 ∈
{1, 2, . . . , p − 1} and a 2 ≡ a 20 mod p. Further P (x) = 2x and P (a 0 ) = 2a 0 = 0
mod p since (a, p) = 1. Therefore by Hensel’s lemma P(x) has a solution in Z p .
Conversely suppose that a is not a quadratic residue. Then P(x) ≡ 0 mod p and
hence P(x) ≡ 0 mod p k for any k. It follows that P(x) can have no solution in Z p .
7.8.1 The Non-isomorphism of the p-Adic Fields
Since each p-adic field is non-archimedean we have seen from the characterization
of R that for any prime p the p-adic field Q p is not isomorphic to R. In the next
theorem we use the results on square roots in Q p to provide another proof of this and
to show that p-adic fields for different primes are non-isomorphic.
Theorem 7.8.4 The p-adic field Q p is not isomorphic to R for any prime p. Further, if p1 and p2 are distinct primes then the corresponding p-adic fields are nonisomorphic.
Proof Let p be a prime and suppose that f : R → Q p is an isomorphism. Then p
has a square root in R and hence by the isomorphism f ( p) has a square root in Q p .
However, p is not a quadratic residue mod p and therefore p has no square root in
Q p providing a contradiction.
If p1 = p2 then there are p12−1 quadratic residues mod p1 and p22−1 quadratic
residues mod p2 . It follows that if p2 > p1 there must exist an integer a which is
7.8 Hensel’s Lemma and Applications
403
a quadratic residue mod p2 but not mod p1 . Use this integer a and then follow the
same proof as above. We leave the details to the exercises.
As a final application of both Hensel’s lemma and the utility of the p-adic fields in
general we mention without proof the local-global principle of Hasse. The rational
numbers Q are called a global field while its Completions, the real numbers R and
the p-adic fields Q p are called local fields. Any relationship among a set of rational
numbers which is true globally, that is in Q is also true locally, that is in R and all
the p-adic fields Q p .
Hasse’s Global-Local Principle provides a partial converse for equations involving quadratic forms with integer coefficients:
ai j xi x j +
i, j
bi xi + c = 0.
i
If such an equation has solutions in R and in Q p for every prime p, then it has a
rational solution in Q. In other words, a quadratic equation with integer coefficients
has a global solution, that is in Q if and only if it has solutions in all the local fields,
that is in R and in Q p for all p.
7.9 Exercises
7.1 Find the p-adic norm and p-adic expansion in Q7 of:
(a) 15
(b) −1
(c) −3
(d) 13
7.2 Describe in detail, analogously as for R, the Cauchy completion of the rational
numbers Q equipped with the p-adic norm for a prime p.
7.3 Fill in the details of the proof of Theorem 7.8.4, that is if p1 = p2 then the
p-adic fields Q p1 and Q p2 are not isomorphic.
7.4 Let p be a prime number and Z p the p-adic integers. Show that Z p / p n Z p is
isomorphic to Z / p n Z for any n > 0.
7.5 Let p be a prime number and Z p the p-adic integers. Show that the additive
group of Z p is torsion-free.
7.6 Use the algorithm in the proof of Hensel’s Lemma to find a solution (if there
exists one) of the polynomial equations:
(a) x 3 − 3x 2 + 2x + 1 = 0 in Q7
(b) x 4 − 6 in Q11
7.7 Complete the proof that a p-adic expansion for x is periodic if and only if x
is rational.
7.8 Show that if x ∈ Q p and x ≡ 0 mod p k for all k ≥ 1 then x = 0.
404
7 The Fields Q p of p-Adic Numbers: Hensel’s Lemma
7.9 Let f (x) ∈ Q p [x] that is a polynomial with p-adic coefficients. Show that
f (x) is a continuous function of Q p .
7.10 Complete the proof of Theorem 7.8.4 and show that if p1 , p2 are distinct
primes then the corresponding p-adic fields are non-isomorphic.
7.11 Prove that the rationals Q are dense in Q p .
7.12 Prove that the p-adic integers Z p are compact as a metric space using the
p-adic norm.
7.13 Show that for any prime p and any positive integer m not divisible by p,
there exists a primitive m-th root of unity in Q p if and only if m divides p − 1.
7.14 Show that the set of roots of unity in Q p is a subgroup of the group of p-adic
units.
7.15 Prove that a rational number x ∈ Q is a square if and only if it is a square in
every p-adic field Q p and in the real numbers R.
7.16 Let Z2 be the 2-adic integers, Show that if b ∈ Z2 and b ≡ 1 mod 8 then b
is a square in Z2 .
7.17 Show that the equation (x 2 − 2)(x 2 − 17)(x 2 − 34) = 0 has a solution in
the real numbers R and in all the p-adic field Q p with p prime, but has no solution
in the rational numbers Q.
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Index
A
Abelian group, 27
ACC, 302
Additive number theory, 4
Affine cipher, 267
AKS algorithm, 239
AKS Algorithm program, 241
Alberti code, 265
Algebraic closure, 316
Algebraic extension, 310
Algebraic integer, 285, 286, 329
Algebraic number, 310, 316
Algebraic number field, 285, 316
Algebraic number theory, 2
Algebraically independent, 201
Analytic continuation, 177
Analytic function, 176
Analytic number theory, 2, 143
Apery’s constant, 173
Archimedean, 381, 384
Arithmetic function, 133
Ascending chain condition, 302
Associates, 21, 287
Asymptotically equal, 154
Attack, 264
Authentication, 270
B
Basis, 309
Beale cipher, 266
Bernoulli numbers, 173, 199
Bertrand’s theorem, 132
Big O notation, 153
Binary expansion, 373
Brun’s constant, 228
© Springer International Publishing AG 2016
B. Fine and G. Rosenberger, Number Theory,
DOI 10.1007/978-3-319-43875-7
C
Caesar code, 265
Canonical p-adic expansion, 390
Carmichael number, 243
Cauchy completion, 382
Cauchy completion procedure, 373
Cauchy integral formula, 176
Cauchy sequence, 374
Cauchy’s Theorem, 176
Chebyshev functions, 160
Cheyshev’s estimate, 147
Chinese Remainder Theorem, 39
Ciphertext message, 263
Class number, 364
Classical cryptography, 264
Common divisor, 11
Commutative ring, 7
Commutative ring with identity, 7
Complete lattice, 342
Complete metric space, 374, 376
Complete residue system, 23
Complex integers, 286
Complext integral, 176
Composite, 11
Congruence, 22
Conjugates, 317
Continued fractions, 110
Convergent sequence, 374
Coprime, 12
Coset of an ideal, 306
Cousin primes, 132, 229
Cramer’s conjecture, 211
Critical line, 183
Critical strip, 183
Cryptanalysis, 264
Cryptography, 219, 263
Cyclic subgroup, 32
409