Editing Local class field theory (section)

==Higher local class field theory==
For a [[Higher local field|higher-dimensional local field]] <math>K</math> there is a higher local reciprocity map which describes abelian extensions of the field in terms of open subgroups of finite index in the [[Milnor K-group]] of the field. Namely, if <math>K</math> is an <math>n</math>-dimensional local field then one uses <math>\mathrm{K}^{\mathrm{M}}_n(K)</math> or its separated quotient endowed with a suitable topology. When <math>n=1</math> the theory becomes the usual local class field theory. Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if <math>n>1</math>. General higher-dimensional local class field theory was developed by [[Kazuya Kato|K. Kato]] and [[Ivan Fesenko|I. Fesenko]].

Higher local class field theory is part of [[higher class field theory]] which studies abelian extensions (resp. abelian covers) of [[Function field (scheme theory)|rational function fields]] of [[proper scheme|proper]] [[Regular scheme|regular]] [[scheme (mathematics)|schemes]] [[flat morphism|flat]] over integers.