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Introduction to Number Theory
Trygve Nagell
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AMS Chelsea Publishing
1964; 309 pp; hardcover
Volume: 163
Reprint/Revision History:
first AMS printing 2001
ISBN-10: 0-8218-2833-9
ISBN-13: 978-0-8218-2833-5
List Price: US$43
Member Price: US$38.70
Order Code: CHEL/163.H
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A special feature of Nagell's well-known text is the rather extensive treatment of Diophantine equations of second and higher degree. A large number of non-routine problems are given.

Reviews

"This is a very readable introduction to number theory, with particular emphasis on diophantine equations, and requires only a school knowledge of mathematics. The exposition is admirably clear. More advanced or recent work is cited as background, where relevant ... [T]here are welcome novelties: Gauss's own evaluation of Gauss's sums, which is still perhaps the most elegant, is reproduced apparently for the first time. There are 180 examples, many of considerable interest, some of these being little known."

-- Mathematical Reviews

Table of Contents

  • Divisibility: 1.1 Divisors; 1.2 Remainders; 1.3 Primes; 1.4 The fundamental theorem; 1.5 Least common multiple and greatest common divisor; 1.6 Moduls, rings and fields; 1.7 Euclid's algorithm; 1.8 Relatively prime numbers. Euler's \(\varphi\)-function; 1.9 Arithmetical functions; 1.10 Diophantine equations of the first degree; 1.11 Lattice points and point lattices; 1.12 Irrational numbers; 1.13 Irrationality of the numbers \(e\) and \(\pi\); Exercises (1--40)
  • On the Distribution of Primes: 2.14 Some lemmata; 2.15 General remarks. The sieve of Eratosthenes; 2.16 The function \(\pi(x)\); 2.17 Some elementary results on the distribution of primes; 2.18 Other problems and results concerning primes
  • Theory of Congruences: 3.19 Definitions and fundamental properties; 3.20 Residue classes and residue systems; 3.21 Fermat's theorem and its generalization by Euler; 3.22 Algebraic congruences and functional congruences; 3.23 Linear congruences; 3.24 Algebraic congruences to a prime modulus; 3.25 Prime divisors of integral polynomials; 3.26 Algebraic congruences to a composite modulus; 3.27 Algebraic congruences to a prime- power modulus; 3.28 Numerical examples of solution of algebraic congruences; 3.29 Divisibility of integral polynomials with regard to a prime modulus; 3.30 Wilson's theorem and its generalization; 3.31 Exponent of an integer modulo \(n\); 3.32 Moduli having primitive roots; 3.33 The index calculus; 3.34 Power residues. Binomial congruences; 3.35 Polynomials representing integers; 3.36 Thue's remainder theorem and its generalization by Scholz; Exercises (41-89)
  • Theory of Quadratic Residues: 4.37 The general quadratic congruence; 4.38 Euler's criterion and Legendre's symbol; 4.39 On the solvability of the congruences \(x^2\equiv\pm 2\, (\text{mod }p)\); 4.40 Gauss's lemma; 4.41 The quadratic reciprocity law; 4.42 Jacobi's symbol and the generalization of the reciprocity law; 4.43 The prime divisors of quadratic polynomials; 4.44 Primes in special arithmetical progressions
  • Arithmetical Properties of the Roots of Unity: 5.45 The roots of unity; 5.46 The cyclotomic polynomial; 5.47 Irreducibility of the cyclotomic polynomial; 5.48 The prime divisors of the cyclotomic polynomial; 5.49 A theorem of Bauer on the prime divisors of certain polynomials; 5.50 On the primes of the form \(n y-1\); 5.51 Some trigonometrical products; 5.52 A polynomial identity of Gauss; 5.53 The Gaussian sums; Exercises (90-122)
  • Diophantine Equations of the Second Degree: 6.54 The representation of integers as sums of integral squares; 6.55 Bachet's theorem; 6.56 The Diophantine equation \(x^2-Dy^2=1\); 6.57 The Diophantine Equation \(x^2-Dy^2=-1\); 6.58 The Diophantine equation \(u^2-Dv^2=C\); 6.59 Lattice points on conics; 6.60 Rational points in the plane and on conics; 6.61 The Diophantine equation \(ax^2+by^2+cz^2=0\)
  • Diophantine Equations of Higher Degree: 7.62 Some Diophantine equations of the fourth degree with three unknowns; 7.63 The Diophantine equation \(2x^4-y^4=z^2\); 7.64 The quadratic fields \(K(\sqrt{-1}), K(\sqrt{-2})\) and \(K(\sqrt{-3})\); 7.65 The Diophantine equation \(\xi^3+\eta^3+\zeta^3=0\) and analogous equations; 7.66 Diophantine equations of the third degree with an infinity of solutions; 7.67 The Diophantine equation \(x^7+y^7+z^7=0\); 7.68 Fermat's last theorem; 7.69 Rational points on plane algebraic curves. Mordell's theorem; 7.70 Lattice points on plane algebraic curves. Theorems of Thue and Siegel; Exercises (123-171)
  • The Prime Number Theorem: 8.71 Lemmata on the order of magnitude of some finite sums; 8.72 Lemmata on the Möbius function and some related functions; 8.73 Further lemmata. Proof of Selberg's formula; 8.74 An elementary proof of the prime number theorem; Exercises (172-180)
  • Table of primitive roots
  • Fundamental solutions of equations \(x^2-Dy^2=\pm 1\)
  • Name index
  • Subject index
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