Tải bản đầy đủ - 0 (trang)
8 Hensel's Lemma and Applications

# 8 Hensel's Lemma and Applications

Tải bản đầy đủ - 0trang

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

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

(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

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.

Bibliography

[AKS]

[AGR]

M. Agrawal, N. Kayal, N. Saxena, PRIMES is in P. Ann. Math. 160(2), 2781–2793 (2004)

W.R. Alford, A. Granville, C. Pomerance, There are infinitely many carmichael numbers.

Ann. Math. 139, 703–722 (1994)

[AG]

I. Anshel, M. Anshel, D. Goldfeld, An algebraic method for public key cryptography.

Math. Res. Lett. 6, 287–291 (1999)

[A]

T.M. Apostol, Introduction to Analytic Number Theory (Springer, New York, 1976)

[Apo]

T.M. Apostol, The Most Surprising Result in Mathematics. The Mathematical Intelligencer (2001)

[Ah]

L. Ahlfors, Introduction to Complex Analysis (Springer, New York, 1976)

[Ba]

A. Baker, Transcendental Number Theory (Cambridge University Press, Cambridge,

1975)

[B]

C.X. Barnes, The infinitude of primes: a proof using continued fractions. L’Enseig. Mathematique 22, 313–316 (1976)

[BH96] R.C. Baker, G. Harmon, The Brun-Titchmarsh theorem on average. Proceedings of a

Conference in Honor of Heini Halberstam 1, 39–103 (1996)

[BFKR] G. Baumslag, B. Fine, M. Kreuzer, G. Rosenberge, A Course in Mathematical Cryptography (DeGruyter, Berlin, 2015)

[BFX] G. Baumslag, B. Fine, X. Xu, Cryptosystems using linear groups, in Proceedings of the

International Conference on Algebraic Cryptography (2005)

[Be]

D. Bernstein, Proving primality after Agrawal, Kayena and Saxena (to appear)

[BK]

M. Berry, J.P. Keating, The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41(2),

236–266 (1999)

[Bo]

F. Bornemann, PRIMES is in P: a breakthrough for everyman. Not. AMS 50(5), 545–552

(2003)

[BFK] J.I. Burgos Gil, J. Fresan, U. Kühn, Classical and motivic multiple zeta values, in Clay

Mathematical Proceedings, in preparation

[Br]

J.W. Bruce, A really trivial proof of the Lucas Lehmer test. Am. Math. Monthly 100,

370–371 (1993)

[C]

E. Cohen, Legendre’s identity. Am. Math. Monthly 76, 611–616 (1969)

[Co]

H. Cohn, A Classical Invitation to Algebraic Numbers and Class Fields (Springer, New

York, 1978)

[CP]

R. Crandall, C. Pomerance, Prime Numbers; A Computational Perspective (Springer,

New York, 2001)

[CR]

C. Curtis, I. Reiner, Representation Theory of Finite Groups (Wiley Interscience, New

York, 1966)

[Da]

H. Davenport, Multiplicative Number Theory (Springer, New York, 1980)

© Springer International Publishing AG 2016

B. Fine and G. Rosenberger, Number Theory,

DOI 10.1007/978-3-319-43875-7

405

406

[DP]

Bibliography

C.J. del la Vallee Poussin, Recherches analytiques sur la theorie des nombres: Premier

partie: La fonction (s) de Riemann et les nombres premiers en general, Annales de la

Soc. scientifique de Bruxelles 20, 183–256 (1896)

[Di]

H.G. Diamond, Elementary methods in the study of the distribution of prime numbers.

Bull. Am. Math. Soc. 7, 553–589 (1982)

[D]

L.E. Dickson, History of the Theory of Numbers (Chelsea, New York, 1950)

[DH]

W. Diffie, M. Hellman, New directions in cryptography. IEEE Trans. Inf. Theory 22,

644–654 (1976)

[E]

P.T.D. Elliott, Probabilistic Number Theory I: Mean Value Theorems (Springer, New

York, 1979)

[E 1]

P.T.D. Elliott, Probabilistic Number Theory II: Central Limit Theorems (Springer, New

York, 1980)

[Er]

P. Erdos, On a new method in elementary number theory which leads to an elementary

proof of the prime number theorem. Proc. Nat. Acad. Sci. USA 35, 374–384 (1949)

[F]

B. Fine, The Algebraic Theory of the Bianchi Groups (Marcel Dekker, New York, 1989)

[F1]

B. Fine, Sums of squares rings. Can. J. Math. 29, 155–160 (1977)

[F2]

B. Fine, A note on the two-square theorem. Can. Math. Bull. 20, 93–94 (1977)

[F3]

B. Fine, Cyclotomic equations and square properties in rings. Int. J. Math. Math. Sci. 9,

89–95 (1986)

[FGR 1] B. Fine, A. Gaglione, G. Rosenberger, Abstract Algebra (Johns Hopkins Press, Baltimore,

2015)

[FR 1] B. Fine, G. Rosenberger, Algebraic Generalizations of Discrete Groups (Marcel Dekker,

New York, 2001)

[FR 2] B. Fine, G. Rosenberger, The Fundamental Theorem of Algebra (Springer, New York,

1999)

[Fou85] E. Fouvry, Theoreme de Brun-Titchmarsh: application au theoreme de Fermat. Invent.

Math. 79, 383–407 (1985)

[Fr]

J. Fraleigh, A First Course in Abstract Algebra, 7th edn. (Addison-Wesley, Reading, 2003)

[Fu]

H. Furstenberg, On the infinitude of primes. Am. Math. Monthy 62, 353 (1955)

[G]

C.F. Gauss, Disquisitiones Arithmeticae, English edn. (Yale University Press, New Haven,

1966)

[Go]

S.W. Golomb, A connected topology for integers. Am. Math. Monthly 24, 663–665 (1966)

[GT]

B. Green, T. Tao, The primes contain arbitarily long arithmetic progressions. Ann. Math.

167, 481–547 (2008)

[Gr]

M.D. Greenberg, Advanced Engineering Mathematics (Prentice Hall, Englewood Cliffs,

1988)

[Ha]

J. Hadamard, Sur la distribution des zeros de la fonction ζ (s) et ses consequences arithmetiques. Bull. de la Soc. Math, de France 24, 199–220 (1896)

[HR]

H. Halberstam, H.E. Richert, Sieve Methods (Academic Press, London, 1974)

[HL]

G.H. Hardy, J.E. Littlewood, A new solution of Waring’s problem. Q. J. Math. 48, 272–293

(1919)

[HW]

G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edn. (Clarendon

Press, Oxford, 1979)

[He]

H.A. Helfgott, The ternary Goldbach conjecture is true, nal (2013). arXiv:1312.7748

[Ho]

P. Hoffman, Archimedes Revenge (Fawcett Crest, New York, 1988)

[J]

D. Johnson, Presentations of Groups (Cambridge University Press, Cambridge, 1990)

[K]

S. Katok, Fuchsian Groups (University of Chicago Press, Chicago, 1992)

[KR 1] G. Kern-Isberner, G. Rosenberger, A note on numbers of the form x 2 + N y 2 . Arch. Math.

43, 148–155 (1984)

[KR 2] G. Kern-Isberner, G. Rosenberger, Normalteiler vom Geschlecht eins in freien Produkten

endlicher zyklischer Gruppen. Results Math. 11, 272–288 (1987)

[Ko]

N. Koblitz, A Course in Number Theory and Cryptography (Springer, New York, 1984)

[L]

E. Landau, Elementary Number Theory (Chelsea, New York, 1958)

Bibliography

407

N. Levinson, More than one third of the zeros of Riemann’s zeta function are on σ = 1/2.

[Lin]

Y.V. Linnik, An elementary solution of Waring’s problem by Shnirelman’s method. Math.

Sbornik 12, 225–230 (1943)

[Li]

J.E. Littlewood, Sur la distribution des nombres premiers. Comptes Rendus Acad. Sci.

Paris 158, 1869–1872 (1914)

[MSU] A.G. Myasnikov, V. Shpilrain, A. Ushakov, Group-Based Cryptography, Advanced

Courses in Mathematics - CRM Barcelona (Birkhäuser, Basel, 2008)

[Ma]

J. Maynard, Small gaps between primes. Ann. Math. 181, 383–413 (2015)

[Mc]

N. McCoy, Elementary Number Theory (Chelsea, New York, 1958)

[N]

M. Nathanson, Elementary Methods in Number Theory (Springer, New York, 2000)

[Na]

W. Narkiewicz, The Development of Prime Number Theory (Springer, New York, 2000)

[Neu]

J. Neukirch, Algebraic Number Theory (Springer, New York, 1999)

[Ne]

D.J. Newman, Simple analytic proof of the prime number theorem. Am. Math. Monthly

87, 693–696 (1980)

[New 1] M. Newman, Integral Matrices (Academic Press, New York, 1972)

[New 2] M. Newman, Matrix representations of groups, National Bureau of Standards (1968)

[NP]

I. Niven, B. Powell, Primes in certain arithmetic progressions. Am. Math. Monthly 83,

467–475 (1976)

[NZ]

I. Niven, H.S. Zuckerman, The Theory of Numbers, 4th edn. (Wiley, New York, 1980)

[NZM] I. Niven, H.S. Zuckerman, H.L. Montgomery, The Theory of Numbers, 5th edn. (Wiley,

New York, 1991)

[O]

O. Ore, Number Theory and Its History (McGraw-Hill, New York, 1949)

[P]

O. Perron, Die Lehre von den Kettenbrücken (Chelsea, New York, 1957)

[PP]

Prime pages. http://primes.utm

[PD]

H. Pollard, H. Diamond, The Theory of Algebraic Numbers, vol. 9 (Carus Mathematical

Monographs (Mathematical Association of America, Washington, 1975)

[Ri]

P. Ribenboim, The Book of Prime Number Records (Springer, New York, 1989)

[Ri 2]

P. Ribenboim, The Little Book of Bigger Primes (Springer, New York, 2004)

[Ri 3]

P. Ribenboim, Die Welt der Primzahlen (Springer, New York, 2011)

[Re]

H.J.J. Te Riele, On the sign of the difference π(x) − Li(x). Math. Comput. 48, 323–328

(1986)

[Rie]

B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebener Groesse Monatsber.

Kgl. Preuss, Akad. Wiss, Berlin, 671–680 (1860)

[RSA] R. Rivest, A. Shamir, L. Adelman, A method for obtaining digital signatures and publickey cryptosystems. Commun. ACM 21, 120–126 (1978)

[Ro]

J. Rotman, The Theory of Groups (W.C. Brown, Dubuque, 1984)

[Sc1]

R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p.

Math. Comput. 44, 483–494 (1985)

[Sc2]

R. Schoof, Counting points on elliptic curves over finite fields. J. Theor. Nombres Bordeaux 7, 219–254 (1995)

[S]

B. Segal, Generalization du theoreme de Brun. Dokl. Akad. Nauk SSSR, 501–507 (1930)

[Se]

A. Selberg, An elementary proof of the prime number theorem. Ann. Math. 50, 305–313

(1949)

[Ser]

J.P. Serre, A Course in Arithmetic (Springer, New York, 1973)

[Sil]

J.H. Silverman, The Arithmetic of Elliptic Curves (Springer, Berlin, 1986)

[St]

D.R. Stinson, Cryptography: Theory and Practice (Chapman and Hall/CRC, Boca Raton,

2002)

[TM]

G. Tenenbaum, M. Mendes-France, The Prime Numbers and Their Distribution, vol. 6

Student Mathemtical Library (American Mathematical Society, Providence, 2000)

[Tu]

J. Tupper, Lucas-Lehmer primality test. http://www.jt-actuary.com/lucas-le.htm

[VC]

J.G. Van der Corput, Ueber Summen von Primzahlen und Primzahlquadraten. Math. Ann.

116, 1–50 (1939)

[Le]

408

Bibliography

[V]

I.M. Vinogradov, On Waring’s Theorem Iz (Akad, Nauk SSSR (English translation in

selected works) (Springer, New York, 1985)

A. Weil, Number Theory; An Approach Through History (Birkhauser, Boston, 1984)

M. Waldschmidt, Lectures on multiple zeta values, IMSC 2011 (2012). http://www.math.

jussieu.fr/~miw/articles/pdf/MZV2011IMSc.pdf

D. Zagier, Die Ersten 50 Millionen Primzahlen (Birkhauser, Boston, 1977)

D. Zagier, Newman’s short proof of the prime number theorem. Am. Math. Monthly 104,

705–708 (1997)

Y. Zhang, Bounded gaps between primes. Ann. Math. 179, 1121–1174 (2014)

V.V. Zudilin, Algebraic relations for multiple zeta values. Uspekhi Mat. Nauk 58, 3–32

(2003)

[W]

[Wa]

[Za]

[Zag]

[Zh]

[Zud]

Index

A

Abelian group, 27

ACC, 302

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

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

### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

8 Hensel's Lemma and Applications

Tải bản đầy đủ ngay(0 tr)

×