Timeline of class field theory

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

• Conrad, Keith, History of class field theory (PDF)
• Hasse, Helmut (1967), "History of class field theory", Algebraic Number Theory, Washington, D.C.: Thompson, pp. 266–279, MR 0218330
• Iyanaga, S. (1975) [1969], "History of class field theory", The theory of numbers, Noth Holland, pp. 479–518
• Roquette, Peter (2001), "Class field theory in characteristic p, its origin and development", Class field theory—its centenary and prospect (Tokyo, 1998), Adv. Stud. Pure Math. 30, Tokyo: Math. Soc. Japan, pp. 549–631,