A Textbook of Algebraic Number Theory

Preface
Acknowledgements
Contents
About the Author
Notation
1 Algebraic Integers, Norm and Trace
	1.1 Historical Background
	1.2 Algebraic Numbers and Algebraic Integers
	1.3 Norm and Trace
2 Integral Basis and Discriminant
	2.1 Notions of Integral Basis and Discriminant
	2.2 Properties of Discriminant
	2.3 Integral Basis and Discriminant of mathbbQ(sqrt[3]m)
	2.4 Integral Basis and Discriminant of Cyclotomic Fields
	2.5 An Algorithm for Computing Integral Basis
3 Properties of the Ring of Algebraic Integers
	3.1 Factorisation into Irreducible Elements
	3.2 mathcalOK as a Dedekind Domain
	3.3 Norm of an Ideal
	3.4 Generalized Fermat's Theorem and Euler's Theorem
	3.5 Characterisation of Imaginary Quadratic Euclidean Fields
4 Splitting of Rational Primes and Dedekind's Theorem
	4.1 Ramification Index and Residual Degree
	4.2 Dedekind's Theorem on Splitting of Primes
	4.3 Splitting of Primes in Quadratic and Cyclotomic Fields
	4.4  Finiteness of Ramified Primes
5 Dirichlet's Unit Theorem
	5.1 Preliminary Results
	5.2 Modification and Application of Minkowski's Lemma on Real Linear Forms
	5.3 Proof of Dirichlet's Unit Theorem
	5.4 Fundamental System of Units and Regulator
	5.5 Computation of Units in Quadratic Fields
6 Prime Ideal Decomposition in Relative Extensions
	6.1 Relative Ramification Index and Residual Degree
	6.2 Splitting of Prime Ideals in Galois Extensions
	6.3 Norm of an Ideal in Relative Extensions
	6.4 The Fundamental Equality in Relative Extensions
7 Relative Discriminant and Dedekind's Theorem on Ramified Primes
	7.1 Notions of Relative Different and Relative Discriminant
	7.2 Relative Discriminant as an Extension of Discriminant
	7.3 Properties of Relative Different and Relative Discriminant
	7.4 Dedekind's Theorem on Ramified Primes
8 Class Group and Class Number
	8.1 Finiteness of Class Number
	8.2 Minkowski's Convex Body Theorem
	8.3 Minkowski's Bound
	8.4 Computation of Class Number
	8.5 Hermite's Theorem on Discriminant
	8.6 A Special Case of Fermat's Last Theorem
9 Dirichlet's Class Number Formula and its Applications
	9.1 Dirichlet's Class Number Formula and Ideal Theorem
	9.2 Proof of Ideal Theorem
	9.3 Derivation of Dirichlet's Class Number Formula
	9.4 Applications of Dirichlet's Class Number Formula
10 Simplified Class Number Formula for Cyclotomic, Quadratic Fields
	10.1 Numerical Characters and L-functions
	10.2 Simplification of Class Number Formula for Cyclotomic Fields
	10.3 Dirichlet's Theorem for Primes in Arithmetic Progressions
	10.4 Jacobi-Kronecker Symbol and Character Associated with a Quadratic Field
	10.5 Simplified Class Number Formula for Quadratic Fields
Appendix A Field Theory
A.1  Introduction
A.2  Algebraic Extensions
A.3  Separable Extensions
A.4  Normal Extensions
A.5  Galois Extensions
A.6  Valued Fields
A.7  Eisenstein-Dumas Irreducibility Criterion
Appendix  Hints and Answers to Selected Exercises
Appendix  References
Index