Local Fields

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

{{For|The concept in mathematics|Local field}}

{{Infobox book

| name = Local Fields

| image = Local Fields - bookcover.jpg

| border = yes

| caption =

| title_orig = Corps Locaux

| genre = Non-fiction

| author = Jean-Pierre Serre

| country = France

| language = French (original)
English (translation)

| subject = Algebraic number theory

| publisher = Springer

| release_date = 1980

| media_type = Print

| pages = 241 pp.

| isbn = 978-0-387-90424-5

| oclc = 4933106

}}

Corps Locaux by Jean-Pierre Serre, originally published in 1962 and translated into English as Local Fields by Marvin Jay Greenberg in 1979, is a seminal graduate-level algebraic number theory text covering local fields, ramification, group cohomology, and 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 field complete with respect to a discrete valuation, but current usage (including later works by Serre) add the condition that the residue class field is finite.{{cite book | last=Cassels | first=J. W. S.| title=Local Fields | publisher=Cambridge University Press | year=1986 | page=v}}

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries,{{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}}

it has over 3500 citations in Google Scholar, and is often referenced with respect.

{{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.{{nLab|id=local+field+%28commutative+algebra%29|title=Local Fields}} the famous text Corps Locaux by Serre{{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}} a classic, beautiful textbook