K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory Second Edition, Springer-Verlag, Berlin, 388 + xiv pp., 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(with integer coe#cients) of the primitive n th roots of unity (that is, the monic polynomial of smallest degree that vanishes on the complex number e 2#i n ) Its degree is Euler s phi function, #(n) the number of positive integers j n that are relatively prime to n. Recall also ([5] prop. 13.2.2) that 1= n L c n (x) where n L means that n is a divisor of L) Since u(x) divides x 1, it must be a product of distinct cyclotomic polynomials, u(x) c n j (x) 4) with n j dividing L. Thus a bound for c n j (2) leads to a bound for # 2 (S) Proposition ....

....c n (2) 1 2 q ) 1 if n has an odd number of distinct positive prime divisors. Proof: Recall the Mobius function , which is defined for positive integers n by (n) 1) if n is squarefree and has k distinct positive prime fa =0otherwise. Following are four basic properties of (see [5]2.2) 1. j) 0ifn 1, 1ifn =1. 2. ab) a) b)ifa and b are relatively prime. 3. j n j(n j ) #(n) 4. c n (x) j n (x . The fourth property is obtained by applying the Mobius inversion formula ( 5] 2.2) to the relation x 1= d n c d (x) The fourth and third ....

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K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory (second edition)", Graduate Texts in Mathematics vol. 84, Springer Verlag, N.Y., 1990.

....Finite Fields This section describes some standard results about quadratic residues and Legendre symbols over finite fields. Readers familiar with this topic can safely skip the next paragraphs and continue with Section 5.6. For more background information one can look up references like [32] or [57]. 5.4 Finite Field Factoids From now on p denotes an odd prime. It is known that there always exists a generator # for the multiplicative group F p k = F p 0 . 32, 57] This means that the sequence #, # , # , will generate all non zero elements of F p k . As this is a set of ....

....the next paragraphs and continue with Section 5.6. For more background information one can look up references like [32] or [57] 5.4 Finite Field Factoids From now on p denotes an odd prime. It is known that there always exists a generator # for the multiplicative group F p k = F p 0 . [32, 57] This means that the sequence #, # , # , will generate all non zero elements of F p k . As this is a set of size p it follows that # = # , and hence # 1) 1. Hence we have the equality if and only if i = j mod (p 1) 5.1) for every integer i and j. We now turn ....

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Kenneth Ireland and Michael Rosen. A Classical Introduction to Modern Number Theory, volume 84 of Graduate Texts in Mathematics. Springer, second edition, 1990.

....theory that this number is given by the multinomial coecient ; jFj ; jFj The next theorem counts T p (A) in case A is an interval. It depends on knowledge of the number of equivalence classes of a certain period. This number is given by the following lemma. See [14] for more information on the classic M obius function. Lemma 6.3. The number of equivalence classes of p periodic elements of F is given by jE p j = 6.6) where is the classic M obius function. Proof. Let us denote the number of p periodic signals in F by N(p) We have that ....

Kenneth Ireland and Michael Rosen. A Classical Introduction to Modern Number Theory, volume 84 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1982.

.... j (ff ) ji ; 0 i 2 Gamma 1: The maps j are called the multiplicative characters of F 2 n : they are homomorphisms from (F 2 n ; Delta) to U . The map 0 is called the trivial multiplicative character. For much more information about characters of finite fields, see [22, 23, 32] Next we define the main class of codes that we ll work with in this paper, the dual BCH codes. In fact, we work with a sub class of these codes, more properly called binary, primitive, dual BCH codes. Let ff be primitive in F 2 n and let t be a positive integer with 1 2t Gamma 1 2 1. ....

K. Ireland and M. Rosen. A Classical Introduction to Modern Number Theory (2nd edition), Graduate Texts in Mathematics Vol. 84. Springer, Berlin, 1990.

....de vista t ecnico no corresponde al nivel de profundidad del curso. La rese na hist orica de la criptograf a cl asica puede verse en el Kahn [Ka96] El texto de Bach y Shallit [BS96] es una excelente referencia para todo lo que concierne a la teor a de n umeros computacional. El Ireland y Rosen [IR90] contiene un profundo tratamiento del area de teor a de n umeros cl asica. El Cormen et al. CLR94] es un texto que puede servir de consulta para todo lo que concierne a algoritmos elementales. Dos buenos textos de consulta para todo lo que tenga que ver con teor a de la complejidad ....

K. Ireland, y M. Rosen, \A classical introduction to modern number theory", Graduate Texts in Mathematics, vol. 84, Springer{Verlag, second edition, 1990.

....GCD computations; assuming that z 1 = z 2 (which can usually be ensured at the expense of 2 units of work) a squaring then requires 12 units and a nonsquaring multiplication requires 9 units of work. The reader who is interested in learning more about the theory of elliptic curves should consult [11], 12] or [15] 5 Lenstra s algorithm The idea of Lenstra s algorithm is to perform a sequence of pseudo random trials, where each trial uses a randomly chosen elliptic curve and has a nonzero probability of finding a factor of N . Let m and m 0 be parameters whose choice will be discussed ....

K. F. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory , SpringerVerlag, 1982, Ch. 18.

.... in Section 3and confirm that such elliptic curves exist in a realistic sense (i.e. constructable) From the point of view of theoretical interest, each construction isd2ELL related to each famous number theory problem: the former is a problem offindGO integer solutions of Pell s equation([16]) and the latter is a problem offind ing twin prime numbers. 4.1 Construction of elliptic curvesredEVOE2 to lower extensiondtens Here we present an algorithm to construct elliptic curves over F p in Corollary 1 since Theorem is a special case of Corollary 1. By using the CMmethodt ] ....

K. Ireland and M. Rosen, A classical introduction to modern number theory, GTM 84, Springer-Verlag, New-York, 1982.

....for any low degree polynomials modulo an odd integer against the parity function, say. The use of exponential sums points to the possibility of applying more sophisticated techniques. There is a strong connection between our sum and the (generalized) Gauss sum or Kloosterman sum (see, e.g. [8]) which we briefly illustrate below. Consider a polynomial f(x 1 ; x n ) with integer coefficients and degree d on n boolean variables. We consider the correlation between, e.g. this polynomial modulo 3 and the parity function Phi. Let be the third root of unity e 2i=3 . Then C n (f; ....

K. Ireland and M. Rosen, A classical introduction to modern number theory, Second Edition, Springer-Verlag, New York, 1990.

....we give the resulting pseudo code and sample C code at the following Web site: http: cristal.inria.fr harley hyper 2 Frobenius Endomorphism In this section we collect some useful results and quote them without proof. A starting point for the reader interested in pursuing this material is [IR82] and the references therein. We first describe properties of the q power Frobenius endomorphism OE(x) x q . Note that it has no effect on elements of F q but it becomes non trivial in 6 Pierrick Gaudry and Robert Harley extension fields. This map extends naturally to points, by transforming ....

K. F. Ireland and M. Rosen. A classical introduction to modern number theory, volume 84 of Graduate texts in Mathematics. Springer--Verlag, 1982.

....curves of order p 1 are explicitly constructable. We will show later that the subgroup of order p 1 of F p 2 works for all p. The following statements about the orders of the curves of the form (3) in the case they are not p 1 can be proved with algebraic number theoretic methods (see [15], 37] If p j 1 (mod 4) p has a unique representation in the ring Z[i] of Gaussian integers: p = a bi) a Gamma bi) a 2 b 2 ; j 1 (mod 2 2i) The curves y 2 = x 3 Gamma Dx have the orders p 1 Sigma 2a; p 1 Sigma 2b; 4) and the four orders occur equally often. ....

K. Ireland and M. Rosen, A classical introduction to modern number theory, Springer-Verlag, 1982.

....12) elliptic curves of order p 1 are explicitly constructable. We will show later that the subgroup of order p 1 of F p 2 is a useful auxiliary group for all p. The following statements about the orders of curves defined by the equations above in the case they are not p 1 are proved in [8]. If p j 1 (mod 4) then p can uniquely be represented as a product in the ring Z[i] of Gaussian integers: p = a bi) a Gamma bi) a 2 b 2 , and j 1 (mod 2 2i) The curves y 2 = x 3 Gamma Dx have the orders p 1 Sigma 2a or p 1 Sigma 2b, and the four orders occur equally ....

K. Ireland and M. Rosen, A classical introduction to modern number theory, Springer-Verlag, 1982.

....congruence fails to be soluble, or the signs of the coefficients are wrong. Indeed, Legendre s own proof follows the same lines: see the account in Weil s historical book [15, p. 100] Algorithmic solutions in the literature often follow essentially the same reduction procedure as Legendre (see [7], or [12] for a recent example) As we will see, the disadvantage of following this in practice is that it takes many steps, each of which involves the factorization of an integer, and the resulting solution can be very large. Our improved method performs much better in all these respects; ....

K. Ireland and M. Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics, no. 84, Springer-Verlag, 1982. 12

....minimal polynomial (with integer coefficients) of the primitive n th roots of unity (that is, the monic polynomial of smallest degree that vanishes on the complex number e 2i=n ) Its degree is Euler s phi function, OE(n) the number of integers j n that are relatively prime to n. Recall also ([5] prop. 13.2.2) that x L Gamma 1 = Y njL c n (x) where njL means that n is a divisor of L) Since u(x) divides x L Gamma 1, it must be a product of distinct cyclotomic polynomials, u(x) t Y j=1 c n j (x) 2) with n j dividing L. Thus a bound for c n j (2) leads to a bound for ....

....2 Gammaq ) Gamma1 if m has an odd number of distinct prime divisors. Proof: Recall the Mobius function , which is defined for positive integers n by (n) Gamma1) k if n is squarefree and has k distinct prime factors, and = 0 otherwise. Following are four basic properties of (see [5]x2.2) 5 1. X jjn (j) 0 if n 1; 1 if n = 1: 2. ab) a) b) if a and b are relatively prime. 3. P jjn j(n=j) OE(n) 4. c n (x) Q jjn (x j Gamma 1) n=j) The fourth property is obtained by applying the Mobius inversion formula ( 5] x2.2) to the relation x n Gamma 1 ....

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K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory (second edition)", Graduate Texts in Mathematics vol. 84, Springer Verlag, N.Y., 1990. 11

.... of computing rst the average over all t is twofold: it encompasses most of the hard work for the case of interest, and it provides a comforting check that the average is 1 for each l, lending support to assumption (ii) We recall the character sum (see, for example, exercise 8 of chapter 5 in [4]) l X t=1 t 2 4p l = 1 if l does not divide 4p. This tells us how many times the Legendre symbol is 1 or 1, given that it is zero for 1 p l values of t (mod l) The prime 2 requires special treatment, as usual. Note that since p is odd, t 2 4p 5 (mod 8) whenever t is odd, ....

K. Ireland and M. Rosen, `A classical introduction to modern number theory', 2nd edition, Springer Graduate Texts in Mathematics 84, (1990)

....congruence fails to be soluble, or the signs of the coefficients are wrong. Indeed, Legendre s own proof follows the same lines: see the account in Weil s historical book [16, p. 100] Algorithmic solutions in the literature often follow essentially the same reduction procedure as Legendre (see [8], or [13] for a recent example) As we will see, this method has two disadvantages in practice: it takes many steps, each of which involves the factorization of an integer, and the resulting solution can be very large. Our first improvement already performs better in these respects; although it ....

K. Ireland and M. Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics, no. 84, Springer-Verlag, 1982.

....a is a square Gamma1; if a is a nonsquare (5) the complex quadratic character of F q . Furthermore, let fx 0 ; x 1 ; x q Gamma1 g be an arbitrary ordering of the elements of F q . By calculation of the Jacobi sum J( X a2Fq (a) 1 Gamma a) Gamma( Gamma1) 6) see Ireland Rosen[2], Theorem 8.3.1c) we get the following orthogonality relation. LEMMA 1 If e i ; i = 0; q, is an orthonormal basis of the (q 1) dimensional linear space over C, then the q 1 vectors a i = 1 p q ( x i x 0 )e 0 (x i x 1 )e 1 : x i x q Gamma1 )e q Gamma1 e q ) i = ....

K. Ireland, M. Rosen, "A Classical Introduction to Modern Number Theory " (Graduate Texts in Mathematics), Springer, Berlin, Heidelberg, New York, 1982.

....proved by Gauss. Theorem 4.4.1 (Law of Quadratic Reciprocity) Let p and q be odd primes. Then (a) 1 p) 1) p 1) 2 , b) 2 p) 1) p 2 1) 8 , c) p q) 1) p 1) 2) q 1) 2) q p) Proof: It is our purpose to give a proof of (c) the proof of (a) and (b) can be found in [68]. Note that q (p q 1 1) There is a primitive q th root of unity in F p q 1 , say #. As # #= 1, # must be a root of (x q 1) x 1) that is, q 1 # i=0 # i = 0. Now let S be the set of quadratic residues in F q and N = F # q S the set of quadratic nonresidues. Then uS = S if u # ....

K. Ireland and R. Rosen, A Classical Introduction to Modern Number Theory, 2nd edition, Graduate Texts in Mathematics V. 84, Springer-Verlag, New York/Heidelberg/Berlin, 1990.

....For its final formulation we need generalized Bernoulli numbers, semilocal) Zeta functions and or L series. Let D = D K=Q 0 be the discriminant K=Q and = K = D : Z Gamma f0; Sigma1g; m 7 ( D m ) Jacobi symbol) be the corresponding multiplicative function (Dirichlet character, see [I R], XVI, x4) factorizing (precisely) through Z=DZ with quadratic rest values ( D p ) at primes p (0 iff pjD) The generalized Bernoulli numbers B n; are defined as coefficients of a power series F (t) 2 Q[ t] namely 1 X n=0 B n; t n n = F (t) jDj X a=1 (a)te at e jDjt ....

....For simplicity we will restrict ourselves to imaginary quadratic number fields K. The Zeta function i K (s) converges absolutely for Re s 1. It has a meromorphic extension to the whole complex plane C with precisely one pole, namely at s = 1, and the pole order there is equal to 1. We refer to [I R], XVI, x6 including the literature given there, to [B S] V, and to [Lan] XIV. The Dirichlet L series of the field K or of the Dirichlet character is defined by L(s; Y p (1 Gamma (p)p Gammas ) Gamma1 = 1 X n=1 (n)n Gammas : 11) It has an analytic extension (without poles) ....

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Ireland,K., Rosen,M., A classical introduction to modern number theory, Grad. Texts in Math. 84, Springer, NY-Heidelberg-Berlin, 1982

.... of computing first the average over all t is twofold: it encompasses most of the hard work for the case of interest, and it provides a comforting check that the average is 1 for each l, lending support to assumption (ii) We recall the character sum (see, for example, exercise 8 of chapter 5 in [4]) l X t=1 t 2 Gamma 4p l = Gamma1 if l does not divide 4p. This tells us how many times the Legendre symbol is 1 or Gamma1, given that it is zero for 1 Gamma p l Delta values of t (mod l) The prime 2 requires special treatment, as usual. Note that since p is odd, t 2 ....

K. Ireland and M. Rosen, `A classical introduction to modern number theory', 2nd edition, Springer Graduate Texts in Mathematics 84, (1990)

....The proof that it respects the equivalence relation is computationally irrelevant, as far as correctness is concerned. Lastly, there is a question as to the computational relevance of the lemma above. It turns out that in the proof we present here of the Chinese remainder theorem, taken from [47], there are both relevant and irrelevant applications of the lemma. I do not see how, in a system based on extraction, one could mark the different instances of the same lemma , in such a way as to reflect the two uses. We turn to the proof of Theorem 4.4.1 above. Proof Firstly, fm 1 ; m n ....

K.Ireland and M.Rosen, A classical introduction to modern number theory, second edition, Springer Graduate Texts in Mathematics no. 84, Springer-Verlag, 1990.

....i; j, and let F = f i;j ) A mod p : 11.1) Here and hereafter, we assume that 0 a mod p p for any integer a. We will compute modulo p the ERD of the matrix F as an auxiliary stage of computing the ERD of A. At first, we should examine if there exists the ERD modulo p of F . Lemma 11.1 [IR82]. Let f(n) be a function defined on the set of positive integers such that f(n) 0 and lim n 1 f(n) 1. Then there exist two positive constants C and n 0 such that, for any n n 0 , the interval J = fp : f(n) n p f(n)g (11.2) contains at least f(n) C log f(n) distinct primes. ....

K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer, Berlin, 1982.

....are both sharp. It is not hard to verify the following: if n = n 0 = p 11 Q 8 i=1 p i , then Q ljn l 1 l Gamma1 log OE(n) if n = p 8 Q 6 i=1 p i , then Q ljn l 1 l Gamma1 log n. 3 Supersingular polynomials In this section, we will quote algebraic number theory from [1] or [3] without comment. Recall that q is a power of the prime p. An algebraic number in C is called a supersingular q number if it is of the form i p q, the product of some root of unity i and the positive square root of q. Obviously it is an algebraic integer. Here we determine all minimal ....

K. Ireland and M. Rosen, A classical introduction to modern number theory, second ed., Graduate Texts in Mathematics, vol. 84, Springer, 1992.

....ffl If for x 2 F p x 3 ax b j 0 mod p, then there are 1 = 0 1 = i x 3 ax b p j 1 value for y in F p for which (2.15) holds, namely y = 0. 2 Next we give examples of curves, where the number of points is easy to compute. For the following four theorems nice proofs can be found in [IrRo90], pp. 304 307. Theorem 2.13 Let p 3 be prime. If p j 3 mod 4, then #E p (a; 0) p 1; 2.47) for all a 2 f1; p Gamma 1g. Theorem 2.14 Let p 3 be prime. If p j 1 mod 4, then: 1. There is a p = fl ffii 2 Z[i] such that p = p p , p j 1 (mod 2 2i) 2. The number of ....

K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1990.

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K. Ireland and M. Rosen, A Classical Introduction to Modern Num ber Theory, 2nd. ed. , Springer, New York (1990).

....de dise no de pol iticas de seguridad en sistemas computacionales. La rese na hist orica de la criptograf ia cl asica puede verse en el Kahn [Ka96] El texto de Bach y Shallit [BS96] es una excelente referencia para todo lo que concierne a la teor ia de n umeros computacional. El Ireland y Rosen [IR90] contiene un profundo tratamiento del area de teor ia de n umeros cl asica. El Cormen et al. CLR94] es un texto que puede servir de consulta para todo lo que concierne a algoritmos elementales. Dos buenos textos de consulta para todo lo que tenga que ver con teor ia de la complejidad ....

K. Ireland, y M. Rosen, "A classical introduction to modern number theory", Graduate Texts in Mathematics, vol. 84, Springer--Verlag, second edition, 1990.

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K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory Second Edition, Springer-Verlag, Berlin, 388 + xiv pp., 1990.

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K. Ireland and M. Rosen, A classical introduction to modern number theory, Grad Texts 84, Springer, 1982.

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K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Second Edition, Springer-Verlag, New York, 1990.

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K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1990.

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K. F. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, Berlin, 1982.

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K. Ireland and M. Rosen, "A classical introduction to modern number theory," in Graduate Texts in Mathematics. New York: Springer-Verlag, 1990, vol. 84.

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K. F. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, Berlin, 1982.

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K. Ireland and M. Rosen. A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics. Springer-Verlag, 2nd edition, 1990.

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K. Ireland and M. Rosen, A classical introduction to modern number theory, Second edition. Graduate Texts in Mathematics, 84. Springer-Verlag, New York, 1990.

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Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed., Springer-Verlag, NY, 1990.

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Ireland, K., Rosen, M. A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1982.

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K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd. ed. , Springer, New York (1990).

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Ireland, K., and Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed. Springer-Verlag, Berlin, 1990.

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K. Ireland, M. Rosen, "A Classical Introduction to Modern Number Theory," Graduate Texts in Math. No. 84. Springer Verlag, Berlin/New York/Heidelberg, 1990.

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K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, SpringerVerlag: New York, 1990.

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K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory", Springer-Verlag, New York, 1982.

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K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer{Verlag, New York, 1982.

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K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer{Verlag, 1982.

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K. Ireland and M. Rosen, A classical introduction to modern number theory, Springer-Verlag, 1982. 29

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K. Ireland, M. Rosen. A classical introduction to modern number theory, GTM 84, Springer-Verlag, New York, 1982

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K. F. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, Berlin, 1982.

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K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, SpringerVerlag: New York, 1990.

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K. F. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, SpringerVerlag, Berlin, 1982.

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K. F. Ireland and M. Rosen. A Classical Introduction to Modern Number Theory. Springer-Verlag, 1982.

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K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer Verlag, N.Y., 1990. 38

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