Author | Ono, Takashi. author |
---|---|
Title | An Introduction to Algebraic Number Theory [electronic resource] / by Takashi Ono |
Imprint | Boston, MA : Springer US, 1990 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-0573-6 |
Descript | XI, 223 p. online resource |
1. To the Gauss Reciprocity Law -- 1.1. Basic Facts -- 1.2. Modules in ? -- 1.3. Euclidean Algorithm and Continued Fractions -- 1.4. Continued-Fraction Expansion of Irrational Numbers -- 1.5. Concept of Groups -- 1.6. Subgroups and Quotient Groups -- 1.7. Ideals and Quotient Rings -- 1.8. Isomorphisms and Homomorphisms -- 1.9. Polynomial Rings -- 1.10. Primitive Roots -- 1.11. Algebraic Integers -- 1.12. Characters of Abelian Groups -- 1.13. The Gauss Reciprocity Law -- 2. Basic Concepts of Algebraic Number Fields -- 2.1. Field Extensions -- 2.2. Galois Theory -- 2.3. Norm, Trace, and Discriminant -- 2.4. Gauss Sum and Jacobi Sum -- 2.5. Construction of a Regular l-gon -- 2.6. Subfields of the lth Cyclotomic Field -- 2.7. Cohomology of Cyclic Groups -- 2.8. Finite Fields -- 2.9. Ring of Integers, Ideals, and Discriminant -- 2.10. Fundamental Theorem of Ideal Theory -- 2.11. Residue Class Rings -- 2.12. Decomposition of Primes in Number Fields -- 2.13. Discriminant and Ramification -- 2.14. Hilbert Theory -- 2.15. Artin Map -- 2.16. Artin Maps of Subfields of the lth Cyclotomic Field -- 2.17. The Artin Map in Quadratic Fields -- 3. Analytic Methods -- 3.1 Lattices in ?n -- 3.2. Minkowskiโs Theorem -- 3.3. Dirichletโs Unit Theorem -- 3.4. Pre-Zeta Functions -- 3.5. Dedekind Zeta Function -- 3.6. The mth Cyclotomic Field -- 3.7. Dirichlet L-Functions -- 3.8. Dirichletโs Theorem on Arithmetical Progressions -- 4. The lth Cyclotomic Field and Quadratic Fields -- 4.1. Determination of Gauss Sums -- 4.2. L-Functions and Gauss Sums -- 4.3. Class Numbers of Subfields of the lth Cyclotomic Field -- 4.4. Class Number of ?$$(\sqrt {{l̂*}} )$$ -- 4.5. Ideal Class Groups of Quadratic Fields -- 4.6. Cohomology of Quadratic Fields -- 4.7. Gauss Genus Theory -- 4.8. Quadratic Irrationals -- 4.9. Real Quadratic Fields and Continued Fractions -- Answers and Hints to Exercises -- Notes -- A. Peano Axioms -- B. Fundamental Theorem of Algebra -- C. Zornโs Lemma -- D. Quadratic Fields and Quadratic Forms -- List of Mathematicians -- Comments on the Bibliography