A Classical Invitation to Algebraic Numbers and Class Fields [Paperback]

Artin's 1932 G?ttingen Lectures on Class Field Theory and Connections between Algebrac Number Theory and Integral Matrices 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 M?bius 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 (Abels 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. Pells equation.- C. Dirichlet theorem.- D. Imbeddings of 0 and 0*.- 12. Geometry of Numbers.- A. Convex bodies.- B. Existence theorem.- C. Parallelopiped applicatiol#7