[PDF] a gentle course in local class field theory pdf

  • What is the course of class field theory?

    Course description
    Class field theory, the study of abelian extensions of number fields, was a crowning achievement of number theory in the first half of the 20th century. It brings together, in a unified fashion, the quadratic and higher reciprocity laws of Gauss, Legendre et al, and vastly generalizes them.

  • What is the main theorem of local class field theory?

    The existence theorem of local class field theory states that the converse also holds. Theorem 27.8 (Local Existence Theorem). Let K be a local field and let H be a finite index open subgroup of K×. There is a unique extension L/K in Kab with NL/K(L×) = H.11 déc. 2017

  • What are the statements of local class field theory?

    Local class field theory gives us canonical bijections between the following sets: (1) finite-index open subgroups of K× (all of which are necessarily normal); (2) open subgroups of Gal(Kab/K) (which are necessarily normal and of finite index); (3) finite abelian extensions of K in Ksep (which necessarily lie in Kab).
  • Class field theory studies finite-dimensional abelian field extensions of number fields and of function fields, hence of global fields by relating them to the idele class group. Class field theory clarifies the origin of various reciprocity laws in number theory.
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cepts in quantum physics and provides a gentle in- troduction to the field of nanotechnology. duce class field theory (local or global) gently? Guillot.



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