A Classical Invitation to Algebraic Numbers and Class Fields (Universitext 1st ed. 1978. 2nd printing 1988)

A Classical Invitation to Algebraic Numbers and Class Fields (Universitext 1st ed. 1978. 2nd printing 1988)

By: Harvey Cohn (author), O. Taussky (assistant)Paperback

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From the reviews/Aus den Besprechungen: "...Fur den an der Geschichte der Zahlentheorie interessierten Mathematikhistoriker ist das Buch mindestens in zweierlei Hinsicht lesenswert. Zum einen enthalt der Text eine ganze Reihe von historischen Hinweisen, zum anderen legt der Autor sehr grossen Wert auf eine moglichst allseitige Motivierung seiner Darlegungen und versucht dazu, insbesondere den wichtigen historischen Schritten auf dem Weg zur Klassenkorpertheorie Rechnung zu tragen. Die Anhange von O. Taussky bilden eine wertvolle Erganzung des Buches. ARTINs Vorlesungen von 1932, deren Ubersetzung auf einem Manuskript basiert, das die Autorin 1932 selbst aus ihrer Vorlesungsnachschrift erarbeitete und von H. HASSE durchgesehen sowie mit Hinweisen versehen wurde, durfte fur Mathematiker und Mathematikhistoriker gleichermassen von Interesse sein..." NTM- Schriftenreihe fur Geschichte der Naturwissenschaften, Technik und Medizin

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Contents

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 Mobius 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 (Gottingen 1932) Notes by O. Taussky.- into Connections Between Algebraic Number Theory and Integral Matrices (Appendix by Olga Taussky).- Subject Matter Index.

Product Details

  • publication date: 25/08/1988
  • ISBN13: 9780387903453
  • Format: Paperback
  • Number Of Pages: 328
  • ID: 9780387903453
  • weight: 539
  • ISBN10: 0387903453
  • edition: 1st ed. 1978. 2nd printing 1988

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