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In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K.Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K.
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.
The origins of class field theory lie in the quadratic reciprocity law proved by Gauss. The generalization took place as a long-term historical project, involving quadratic forms and their 'genus theory', work of Ernst Kummer and Leopold Kronecker/Kurt Hensel on ideals and completions, the theory of cyclotomic and Kummer extensions.
The principal ideal theorem was conjectured by David Hilbert (), and was the last remaining aspect of his program on class fields to be completed, in 1929.. Emil Artin (1927, 1929) reduced the principal ideal theorem to a question about finite abelian groups: he showed that it would follow if the transfer from a finite group to its derived subgroup is trivial.
Corry (1996) and Schappacher (2005) and the English introduction to (Hilbert 1998) give detailed discussions of the history and influence of Hilbert's Zahlbericht. Some earlier reports on number theory include the report by H. J. S. Smith in 6 parts between 1859 and 1865, reprinted in Smith (1965), and the report by Brill & Noether (1894).
Hilbert field may refer to: The Hilbert field, the minimal ordered Pythagorean field; A Hilbert field is one with minimal Kaplansky radical; Hilbert class field, the maximal abelian unramified extension of a number field; Hilbert–Speiser field, a field with a normal integral basis
Here we are using Hilbert series of filtered algebras, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra. Thus R 0 {\displaystyle R_{0}} is an Artinian ring , which is a k -vector space of dimension P (1) , and Jordan–Hölder theorem may be used for proving that P (1) is the degree of the ...
Hilbert, the first of two children and only son of Otto, a county judge, and Maria Therese Hilbert (née Erdtmann), the daughter of a merchant, was born in the Province of Prussia, Kingdom of Prussia, either in Königsberg (according to Hilbert's own statement) or in Wehlau (known since 1946 as Znamensk) near Königsberg where his father worked ...