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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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Timeline of class field theory」の詳細全文を読む スポンサード リンク
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