Local class field theory

In mathematics, local class field theory, introduced by Helmut Hasse, 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 Q_p (where p is any prime number), or a finite extension of the field of formal Laurent series F_q((T)) over a finite field F_q.

Local class field theory gives a description of the Galois group G of the maximal abelian extension of a local field K via the reciprocity map which acts from the multiplicative group K^×=K\{0}. For a finite abelian extension L of K the reciprocity map induces an isomorphism of the quotient group K^×/N(L^×) of K^× by the norm group N(L^×) of the extension L^× to the Galois group Gal(L/K) of the extension.

The existence theorem in local class field theory establishes a one-to-one correspondence between open subgroups of finite index in the multiplicative group K^× and finite abelian extensions of the field K. For a finite abelian extension L of K the corresponding open subgroup of finite index is the noram group N(L^×). The reciprocity amp sends higher groups of units to higher ramification subgroups, see e.g. Ch. IV of.

Using the local reciprocity map, one defines the Hilbert symbol and its generalizations. Finding explicit formulas for it is one of subdirections of the theory of local fields, it has a long and rich history, see e.g. Sergei Vostokov's review.

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