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Local class field theory
In mathematics, local class field theory, introduced by Helmut Hasse,[1] is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbers Qp (where p is any prime number), or a finite extension of the field of formal Laurent series Fq((T)) over a finite field Fq.
Generalizations of local class field theory
Generalizations of local class field theory to local fields with quasi-finite residue field were easy extensions of the theory, obtained by G. Whaples in the 1950s, see chapter V of[clarification needed].[6]
Explicit p-class field theory for local fields with perfect and imperfect residue fields which are not finite has to deal with the new issue of norm groups of infinite index. Appropriate theories were constructed by Ivan Fesenko.[7][8] Fesenko's noncommutative local class field theory for arithmetically profinite Galois extensions of local fields studies appropriate local reciprocity cocycle map and its properties.[9] This arithmetic theory can be viewed as an alternative to the representation theoretical local Langlands correspondence.
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