A Textbook of Algebraic Number Theory

Description

This self-contained and comprehensive textbook of algebraic number theory is useful for advanced undergraduate and graduate students of mathematics. The book discusses proofs of almost all basic significant theorems of algebraic number theory including Dedekind’s theorem on splitting of primes, Dirichlet’s unit theorem, Minkowski’s convex body theorem, Dedekind’s discriminant theorem, Hermite’s theorem on discriminant, Dirichlet’s class number formula, and Dirichlet’s theorem on primes in arithmetic progressions. A few research problems arising out of these results are mentioned together with the progress made in the direction of each problem. Following the classical approach of Dedekind’s theory of ideals, the book aims at arousing the reader’s interest in the current research being held in the subject area. It not only proves basic results but pairs them with recent developments, making the book relevant and thought-provoking. Historical notes are given at various places. Featured with numerous related exercises and examples, this book is of significant value to students and researchers associated with the field. The book also is suitable for independent study. The only prerequisite is basic knowledge of abstract algebra and elementary 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