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8 Hensel's Lemma and Applications

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

8 Hensel's Lemma and Applications