Supriya Ghosh (Editor)

Timeline of class field theory

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In mathematics, class field theory is the study of abelian extensions of local and global fields.

Timeline

  • 1801 Gauss proves the law of quadratic reciprocity
  • 1829 Abel uses special values of the lemniscate function to construct abelian extensions of Q(i).
  • 1837 Dirichlet's theorem on arithmetic progressions.
  • 1853 Kronecker announces the Kronecker–Weber theorem
  • 1880 Kronecker introduces his Jugendtraum about abelian extensions of imaginary quadratic fields
  • 1886 Weber proves the Kronecker–Weber theorem (with a slight gap)
  • 1896 Hilbert gives the first complete proof of the Kronecker–Weber theorem
  • 1897 Weber introduces ray class groups and general ideal class groups
  • 1897 Hilbert publishes his Zahlbericht.
  • 1897 Hilbert rewrites the law of quadratic reciprocity as a product formula for the Hilbert symbol.
  • 1897 Hensel introduced p-adic numbers
  • 1898 Hilbert conjectures the existence and properties of the (narrow) Hilbert class field, proving them in the special case of class number 2.
  • 1907 Furtwangler proves existence and basic properties of the Hilbert class field
  • 1908 Weber defines the class field of a general ideal class group
  • 1920 Takagi shows that the abelian extensions of a number field are exactly the class fields of ideal class groups.
  • 1922 Takagi's paper on reciprocity laws
  • 1923 Hasse introduced the Hasse principle (for the special case of quadratic forms).
  • 1923 Artin conjectures his reciprocity law
  • 1924 Artin introduces Artin L-functions
  • 1926 Chebotarev proves his density theorem
  • 1927 Artin proves his reciprocity law giving a canonical isomorphism between Galois groups and ideal class groups
  • 1930 Furtwangler and Artin prove the principal ideal theorem
  • 1930 Hasse introduces local class field theory
  • 1931 Hasse proves the Hasse norm theorem
  • 1931 Hasse classifies simple algebras over local fields
  • 1931 Herbrand introduces the Herbrand quotient.
  • 1931 The Brauer-Hasse-Noether theorem proves the Hasse principle for simple algebras over global fields.
  • 1933 Hasse classifies simple algebras over number fields
  • 1934 Deuring and Noether develop class field theory using algebras
  • 1936 Chevalley introduces ideles
  • 1940 Chevalley uses ideles to give an algebraic proof of the second inequality for abelian extensions
  • 1948 Wang proves the Grunwald–Wang theorem, correcting an error of Grunwald's.
  • 1950 Tate's thesis uses analysis on adele rings to study zeta functions
  • 1951 Weil introduces Weil groups
  • 1952 Artin and Tate introduce class formations in their notes on class field theory
  • 1952 Hochschild and Nakayama introduce group cohomology into class field theory
  • 1952 Tate introduces Tate cohomology groups
  • 1964 Golod and Shafarevich prove that the class field tower can be infinite
  • 1965 Lubin and Tate use Lubin–Tate formal group laws to construct ramified abelian extensions of local fields.

    • References

    Timeline of class field theory Wikipedia
    (Text) CC BY-SA