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

Timeline

• 1801 Carl Friedrich Gauss proves the law of quadratic reciprocity
• 1829 Niels Henrik Abel uses special values of the lemniscate function to construct abelian extensions of ${\displaystyle \mathbb {Q} (i)}$.
• 1837 Dirichlet's theorem on arithmetic progressions.
• 1853 Leopold Kronecker announces the Kronecker–Weber theorem
• 1880 Kronecker introduces his Jugendtraum about abelian extensions of imaginary quadratic fields
• 1886 Heinrich Martin Weber proves the Kronecker–Weber theorem (with a slight gap).
• 1896 David 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 Kurt 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 Philipp Furtwängler proves existence and basic properties of the Hilbert class field.
• 1908 Weber defines the class field of a general ideal class group.
• 1920 Teiji 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 Helmut Hasse introduced the Hasse principle (for the special case of quadratic forms).
• 1923 Emil Artin conjectures his reciprocity law.
• 1924 Artin introduces Artin L-functions.
• 1926 Nikolai Chebotaryov proves his density theorem.
• 1927 Artin proves his reciprocity law giving a canonical isomorphism between Galois groups and ideal class groups.
• 1930 Furtwängler 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 Jacques Herbrand introduces the Herbrand quotient.
• 1931 The Albert–Brauer–Hasse–Noether theorem proves the Hasse principle for simple algebras over global fields.
• 1933 Hasse classifies simple algebras over number fields.
• 1934 Max Deuring and Emmy Noether develop class field theory using algebras.
• 1936 Claude Chevalley introduces ideles.
• 1940 Chevalley uses ideles to give an algebraic proof of the second inequality for abelian extensions.
• 1948 Shianghao 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 André Weil introduces Weil groups.
• 1952 Artin and Tate introduce class formations in their notes on class field theory.
• 1952 Gerhard Hochschild and Tadashi Nakayama introduce group cohomology into class field theory.
• 1952 John Tate introduces Tate cohomology groups.
• 1964 Evgeny Golod and Igor Shafarevich prove that the class field tower can be infinite.
• 1965 Jonathan 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)
• Fesenko, Ivan, Class field theory, its three main generalisations, and applications, EMS Surveys in Mathematical Sciences 2021
• 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, North 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., vol. 30, Tokyo: Math. Soc. Japan, pp. 549–631