Local Fields

For The concept in mathematics, see Local field.

Local Fields

Author	Jean-Pierre Serre
Original title	Corps Locaux
Language	French (original) English (translation)
Subject	Algebraic number theory
Genre	Non-fiction
Publisher	Springer
Publication date	1980
Publication place	France
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 a 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. ^[1]

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries. ^[2] It has over 3500 citations in Google Scholar, and is often referenced with respect. ^{[3] [4] [5]}

Contents

1. Part I, Local Fields (Basic Facts): Discrete valuation rings, Dedekind domains, and Completion.
2. Part II, Ramification: Discriminant & Different, Ramification Groups, The Norm, and Artin Representation.
3. 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.
4. Part IV, Local Class Field Theory: Brauer Group of a Local Field, Local Class Field Theory, Local Symbols and Existence Theorem, and Ramification.

Citations

1. Cassels, J. W. S. (1986). Local Fields. Cambridge University Press. p. v.
2. Berg, Michael (2020-02-23). "Local Fields" (PDF). MAA Reviews. Retrieved 2025-05-30.
3. Raskin, Sam (2016). "Number Theory II: Class Field Theory". MIT OpenCourseWare Reviews. Retrieved 2025-05-30.A classic reference that rewards the effort you put into it.
4. Local Fields at the nLab the famous text Corps Locaux by Serre
5. Guillot, Pierre (2018). A gentle course in local class field theory: local number fields, Brauer groups, Galois cohomology. Cambridge University Press.a classic, beautiful textbook

References

• Serre, Jean-Pierre (1980), Local Fields, Berlin, New York: Springer-Verlag, ISBN   978-0-387-90424-5, MR   0554237