A Classical Invitation to Algebraic Numbers and Class Fields

I. Preliminaries.- 1. Introductory Remarks on Quadratic Forms.- 2. Algebraic Background.- A. Factorial rings (ufd).- B. Integral elements.- C. Euclidean domains.- D. Modules and ideals.- E. Principal ideal domains (pid).- F. Rational integers.- 3. Quadratic Euclidean Rings.- 4. Congruence Classes.- A. Norm and phi-function.- B. Module operations.- C. Chinese remainder theorem.- D. Euler phi-function and Möbius mu-function.- E. Rational residue class groups.- F. Quadratic reciprocity.- 5. Polynomial Rings.- A. Factorization properties.- B. Finite fields.- C. Abstract model and automorphisms.- 6. Dedekind Domains.- A. Prime and maximal ideals.- B. Noether axioms.- C. Sufficiency of axioms.- D. Equivalence classes.- 7. Extensions of Dedekind Domains.- A. Validity of axioms.- B. Root-discriminant.- C. Basis of theorems of Hermite and Smith.- 8. Rational and Elliptic Functions.- A. Rational functions.- B. Elliptic functions.- C. Riemann surfaces.- D. Ideal structure.- E. Principal ideals (Abel’s theorem).- II. Ideal Structure in Number Fields.- 9. Basis and Discriminant.- A. Free nonsingular basis.- B. Norm and trace.- C. Conjugates.- D. Basis and discriminant computation.- E. Quadratic field $$\Phi \left( {\sqrt D } \right)$$.- F. Pure cubic field $$\Phi \left( {\sqrt[3]{m}} \right) $$.- G. Cyclotomic field $$\Phi \left( {\exp 2\pi i/m} \right)$$.- H. Ring index.- 10. Prime Factorization.- A. Main theorem.- B. Ring ideals.- C. Quadratic field $$\Phi \left( {\sqrt m } \right)$$.- D. Kronecker symbol.- E. Pure cubic field $$\Phi \left( {\sqrt[3]{m}} \right)$$.- F. Cyclotomic field $$\Phi \left( {\exp 2\pi i/m} \right)$$.- G. Discriminantal divisors.- 11. Units.- A. Quadratic fields.- B. Pell’s equation.- C. Dirichlet theorem.- D. Imbeddings of 0 and 0*.- 12. Geometry of Numbers.- A. Convex bodies.- B. Existence theorem.- C. Parallelopiped applications.- D. Octahedron (norm) applications.- E. Volume coordinates.- 13. Finite Determination of Class Number.- A. Primary associates.- B. Norm estimates and class number.- C. Norm density.- D. Zeta function.- E. Quadratic case.- III. Introduction to Class Field Theory.- 14. Quadratic Forms, Rings and Genera.- A. Forms and modules.- B. Strict equivalence.- C. Ring equivalence.- D. Genus equivalence.- E. Number of genera.- F. Quadratic reciprocity.- G. Genus characters.- H. p-adic numbers.- I. Norm-residue theory: Hilbert symbol.- 15. Ray Class Structure and Fields, Hilbert Class Fields.- A. Ray modulus semigroup.- B. Ray number groups.- C. Ray ideal groups.- D. Conductor and maximal ray ideal group.- E. Weber-Takagi correspondence.- F. Rational base field.- G. Genus extension field.- H. Hilbert class field.- I. Ring class fields.- 16. Hilbert Sequences.- A. Galois groups.- B. Classical examples.- C. Relative norms.- D. Definition of Hilbert sequence.- E. Illustrations (and quadratic reciprocity again).- F. Tchebotareff monodromy theorem.- 17 Discriminant and Conductor.- A. Relative different and discriminant.- B. Kronecker’s theory of forms.- C. Hensel’s local theory.- D. Relative quadratic fields.- E. Ramification in Hilbert sequence.- F. Conductor-discriminant relation.- G. Relative bases.- 18. The Artin Isomorphism.- A. Artin symbol.- B. Illustrations.- C. Artin reciprocity.- D. Automorphisms of base fields.- E. Arithmetic invariants.- F. Group extensions and class field transfers.- G. Dirichlet genus characters.- 19. The Zeta-Function.- A. Class number relations.- B. Unit relations.- C. Hecke L-function.- D. Tchebotareff density theorem.- E. Analytic motivation of class field.- F. Artin L-function.- Appendices (by Olga Taussky).- Lectures on Class Field Theory by E. Artin (Göttingen 1932) Notes by O. Taussky.- into Connections Between Algebraic Number Theory and Integral Matrices (Appendix by Olga Taussky).- Subject Matter Index.