# Local Fields

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{{short description|Book by Jean-Pierre Serre}}

{{For|the concept in mathematics|Local field}}
{{Infobox book<!-- See Wikipedia:WikiProject_Novels or Wikipedia:WikiProject_Books -->
| name          = Local Fields
| image         = Local Fields - bookcover.jpg
| border        = yes
| caption       =
| title_orig    = Corps Locaux
| genre         = Non-fiction
| author        = [Jean-Pierre Serre](/source/Jean-Pierre_Serre)
| country       = France
| language      = [French](/source/French_language) (original)<br /> [English](/source/English_language) (translation)
| subject       = [Algebraic number theory](/source/Algebraic_number_theory)
| publisher     = [Springer](/source/Springer_Science%2BBusiness_Media)
| release_date  = 1980
| media_type    = Print
| pages         = 241 pp.
| isbn          = 978-0-387-90424-5
| oclc          = 4933106
}}

'''''Corps Locaux''''' by [Jean-Pierre Serre](/source/Jean-Pierre_Serre), originally published in 1962 and translated into English as '''''Local Fields''''' by [Marvin Jay Greenberg](/source/Marvin_Jay_Greenberg) in 1979, is a seminal graduate-level [algebraic number theory](/source/algebraic_number_theory) text covering [local field](/source/local_field)s, [ramification](/source/ramification_(mathematics)), [group cohomology](/source/group_cohomology), and [local class field theory](/source/local_class_field_theory).  The book's end goal is to present local class field theory from the cohomological point of view. 

In this book, a "local field" is defined as a [field](/source/field_(mathematics)) [complete](/source/complete_metric_space) with respect to a [discrete valuation](/source/discrete_valuation), but current usage (including later works by Serre) add the condition that the [residue class field](/source/residue_field) is [finite](/source/finite_field).<ref>{{cite book | last=Cassels | first=J. W. S. |author-link=J. W. S. Cassels| title=Local Fields  | publisher=[Cambridge University Press](/source/Cambridge_University_Press)  | year=1986 | page=v}}</ref>

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.<ref>{{cite web |last= Berg|first= Michael |date=2020-02-23 |title= Local Fields|url= https://old.maa.org/press/maa-reviews/local-fields |website=MAA Reviews|access-date=2025-05-30|format=PDF}}</ref> It has over 3500 citations in [Google Scholar](/source/Google_Scholar), and is often referenced with respect.<ref>{{cite web |last= Raskin|first= Sam |date=2016 |title= Number Theory II: Class Field Theory|url= https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/pages/readings/ |website=MIT OpenCourseWare Reviews|access-date=2025-05-30}} ''A classic reference that rewards the effort you put into it.''</ref><ref>{{nLab|id=local+field+%28commutative+algebra%29|title=Local Fields}} ''the famous text Corps Locaux by Serre''</ref><ref>{{cite book |last= Guillot|first=Pierre |date=2018 |title= A gentle course in local class field theory: local number fields, Brauer groups, Galois cohomology|publisher=[Cambridge University Press](/source/Cambridge_University_Press)}} ''a classic, beautiful textbook''</ref>

==Contents==
#''Part I, Local Fields (Basic Facts)'': Discrete valuation rings, Dedekind domains, and Completion.
#''Part II, Ramification'': Discriminant & Different, Ramification Groups, The Norm, and Artin Representation.
#''Part III, Group Cohomology'': Abelian & Nonabelian Cohomology, Cohomology of Finite Groups, Theorems of Tate and Nakayama, Galois Cohomology, Class Formations, and Computation of Cup Products.
#''Part IV, Local Class Field Theory'': Brauer Group of a Local Field, Local Class Field Theory, Local Symbols and Existence Theorem, and Ramification.

==Citations==
{{reflist}}

==References==
*{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Local Fields  | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | isbn=978-0-387-90424-5 |mr=0554237 | year=1980}}

Category:Algebraic number theory
Category:Class field theory
Category:Graduate Texts in Mathematics
Category:Mathematics textbooks

{{mathematics-lit-stub}}

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Adapted from the Wikipedia article [Local Fields](https://en.wikipedia.org/wiki/Local_Fields) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Local_Fields?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.