The elements of the theory of algebraic numbers

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Title page Preface ERRATA INTRODUCTION CHAPTER I. PRELIMINARY DEFINITIONS AND THEOREMS 1. Algebraic numbers. Algebraic integers. Degree of an algebraic number 2. Algebraic number realms 3. Generation of a realm 4. Degree of a realm. Conjugate realm. Conjugate numbers 5. Forecast of remaining chapters CHAPTER II. THE RATIONAL REALM. Divisibility of Integers 1. Numbers of the rational realm 2. Integers of the rational realm 3. Definition of divisibility 4. Units of the rational realm 5. Rational prime numbers 6. The rational primes are infinite in number 7. Unique factorization theorem 8. Divisors of an integer 9. Determination of the highest power of a prime, p, by which m! is divisible 10. The quotient m/(a!b!...k!) where m = a+b+...+k, is an integer CHAPTER III. THE RATIONAL REALM. Congruences 1. Definition. Elementary theorems 2. The function φ(m) 3. The product theorem for the q,-function 4. The summation theorem for the φ-function 5. Discussion of certain functional equations and another derivation of the general expression of φ(m) in terms of m 6. φ-functions of higher order 7. Residue systems formed by multiplying the numbers of a given system by an integer prime to the modulus 8. Fermat's Theorem as generalized by Euler 9. Congruences of condition. Preliminary discussion 10. Equivalent congruences 11. Systems of congruences. Equivalent systems 12. Congruences in one unknown. Comparison with equations 13. Congruences of the first degree in one unknown 14. Determination of an integer that has certain residues with respect to a given series of moduli 15. Divisibility of one polynomial by another with respect to a prime modulus. Common divisors. Common multiples 16. Unit and associated polynomials with respect to a prime modulus. Primary polynomials 17. Prime polynomials with respect to a prime modulus. Determination of the prime polynomials, mod p, of any given degree 18. Division of one polynomial by another with respect to a prime modulus 19. Congruence of two polynomials with respect to a double modulus 20. Unique fractionization theorem for polynomials with respect to a prime modulus 21. Resolution of a polynomial into its prime factors with respect to a prime modulus 22. The general congruence of the nth degree in one unknown and with prime modulus 23. The congruence x^{φ(m)} - 1 = 0, mod m 24. Wilson's Theorern 25. Common roots of two congruences 26. Determination of the multiple roots of a congruence with prime modulus 27. Congruences in one unknown and with composite modulus 28. Residues of powers 29. Primitive roots 30. Indices 31. Solution of congruences by means of indices 32. Binomial congruences 33. Determination of a primitive root of a given prime number 34. The congruence x^n = b, mod p. Euler's criterion CHAPTER IV. THE RATIONAL REALM. Quadratic Residues 1. The general congruence of the second degree with one unknown 2. Quadratic residues and non-residues 3. Determination of the quadratic residues and non-residues of a given odd prime modulus 4. Legendre's Symbol 5. Determination of the odd prime moduli of which a given integer is a quadratic residue 6. Prime moduli of which - 1 is a quadratic residue 7. Determination of a root of the congruence x² = - l, mod p, (p = 4m+1) by means of Wilson's Theorem 8. Gauss' Lemma 9. Prime moduli of which 2 is a quadratic residue 10. Law of reciprocity for quadratic residues 11. Determination of the value of (a/p) by means of the quadratic reciprocity law, a being any given integer and p a prime 12. Determination of the odd prime moduli of which a given positive odd prime is a quadratic residue 13. Determination of the odd prime moduli of which any given integer is a quadratic residue 14. Other applications of the quadratic reciprocity law CHAPTER V. THE REALM k(i) 1. Numbers of k(i). Conjugate and norm of a number 2. Integers of k(i) 3. Basis of k(i) 4. Discriminant of k(i) 5. Divisibility of integers of k(i) 6. Units of k(i). Associated integers 7. Prime numbers of k(i) 8. Unique factorization theorem for k(i) 9. Classification of the prime numbers of k(i) 10. Factorization of a rational prime in k(i) determined by the value of (d/p) 11 Congruences in k(i) 12. The φ-function in k(i) 13. Residue systems formed by multiplying the numbers of a given system by an integer prime to the modulus 14. The analogue for k(i) of Fermat's Theorem 15. Congruences of condition 16. Two problems 17. Primary integers of k(i) 18. Quadratic residues and the quadratic reciprocity law in k(i) 19. Biquadratic residues CHAPTER VI. THE REALM k(√-3) 1. Numbers of k(√-3) 2. Integers of k(√-3) 3. Basis of k(√-3) 4. Conjugate and norm of an integer of k(√-3) 5. Discriminant of k(√-3) 6. Divisibility of integers of k(√-3) 7. Units of k(√-3). Associated integers 8. Prime numbers of k(√-3) 9. Unique factorization theorem for k(√-3) 10. Classification of the prime numbers of k(√-3) 11. Factorization of a rational prime in k(√-3) determined by the value of (d/p) 12. Cubic residues CHAPTER VII. THE REALM k(√2) 1. Numbers of k(√2) 2. Integers of k(√2) 3. Discriminant of k(√2) 4. Divisibility of integers of k(√2) 5. Units of k(√2). Associated integers 6. Prime numbers of k(√2) 7. Unique factorization theorem for k(√2) 8. Classification of the prime numbers of k(√2) 9. Factorization of a rational prime in k(√2) determined by the value of (d/p) 10. Congruences in k(√2) 11. The Diophantine equations x² - 2y² = +-1, x² - 2y² = +-p, and x² - 2y² = +-m CHAPTER VIII. THE REALM k(√-5) 1. Numbers of k(√-5) 2. Integers of k(√-5) 3. Discriminant of (√-5) 4. Divisibility of integers of k(√-5) 5. Units of k(√-5) . Associated integers 6. Prime numbers of k(√-5) 7. Failure of the unique factorization theorem in k(√-5). Introduction of the ideal 8. Definition of an ideal of k(√-5) 9. Equality of ideals 10. Principal and non-principal ideals 11. Multiplication of ideals 12. Divisibility of ideals 13. The unit ideal 14. Prime ideals 15. Restoration of the unique factorization law in terms of ideal factors CHAPTER IX. GENERAL THEOREMS CONCERNING ALGEBRAIC NUMBERS 1. Polynomials in a single variable 2. Numbers of a realm CHAPTER X. THE GENERAL QUADRATIC REALM 1. Number defining the realm 2. Numbers of the realm. Conjugate and norm of a number. Primitive and imprimitive numbers 3. Discriminant of a nnmber 4. Basis of a quadratic realm 5. Discriminant of the realm 6. Determination of a basis of k(√m) CHAPTER XI. IDEALS OF A QUADRATC REALM 1. Definition. Numbers of an ideal 2. Basis of an ideal. Canonical basis. Principal and non-principal ideals 3. Conjugate of an ideal 4- Equality of ideals 5. Multiplication of ideals 6. Divisibility of ideals. The unit ideal. Prime ideals 7. Unique factorization theorem for ideals CHAPTER XII. CONGRUENCES WHOSE MODULI ARE IDEALS 1. Definition. Elementary theorems 2. The norm of an ideal. Classification of the numbers of an ideal with respect to another ideal 3. Determination and classification of the prime ideals of a quadratic realm 4. Resolution of any given ideal into its prime factors 5. Determination of the norm of any given ideal 6. Determination of a basis of any given ideal 7. Determination of a number α of any ideal a such that (α)/a is prime to a given ideal, m 8. The φ-function for ideals 9. Residue systems formed by multiplying the numbers of a given system by an integer prime to the modulus 10. The analogue for ideals of Fermat's Theorem 11. Congruences of condition 12. Equivalent congruences 13. Congruences in one unknown with ideal moduli 14. The general congruence of first degree with one unknown 15. Divisibility of one polynomial by another with respect to a prime ideal modulus. Common divisors. Common multiples 16. Unit and associated polynomials with respect to a prime ideal modulus. Primary polynomials 17. Prime polynomials with respect to a prime ideal modulus. Determination of the prime polynomials, mod p, of any given degree 18. Division of one polynomial by another with respect to a prime ideal modulus 19. Unique fractorization theorem for polynomials with respect to a prime ideal modulus 20. The general congruence of the nth degree in one unknown and with prime ideal modulus 21. The congruence x^{φ(m)}- 1 = 0, mod m 22. The analogue for ideals of Wilson's Theorem 23. Common roots of two congruences 24. Determination of the multiple roots of a congruence with prime ideal modulus 25. Solution of congruences in one unknown and with composite modulus 26. Residues of powers for ideal moduli 27. Primitive numbers with respect to a prime ideal modulus 28. Indices 29. Solution of congruences by means of indices CHAPTER XIII. THE UNITS oF THE GENERAL QUADRATIC REALM 1. Definition 2. Units of an imaginary quadratic realm 3. Units of a real quadratic realm 4. Determination of the fundamental unit 5. Pell's Equation CHAPTER XIV. THE IDEAL CLASSES OF A QUADRATIC REALM 1. Equivalence of ideals 2. Ideal classes 3. The class number of a quadratic realm Index