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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.
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).
As with the Hilbert problems, one of the prize problems (the Poincaré conjecture) was solved relatively soon after the problems were announced. The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise.
The use of "Hilbert-style" and similar terms to describe axiomatic proof systems in logic is due to the influence of Hilbert and Ackermann's Principles of Mathematical Logic (1928). [2] Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference.
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.
A worker clears snow off school buses, after schools were closed due to a storm, in Manchester, N.H., Monday, Feb. 13, 2017. Another winter blast of snow and strong winds moved into the Northeast ...
The following have announced closings or cancellations because of the storm that is expected to bring heavy rain and strong winds that could develop into tornadoes.
Myrtle Beach, SC area closings and cancellations due to Helene. Terri Richardson. September 26, 2024 at 1:00 AM. 1 / 2. ... extensions to the due date will be announced, the post said.