Corps Locaux

Corps Locaux (Local Fields) is a mathematics textbook published in 1962 based on lectures given by Jean-Pierre Serre at the Collège de France. The text presents the theory of local fields, which are fundamental objects in algebraic number theory and arithmetic geometry. The book develops the core concepts starting from basic field theory and progresses through local field properties, extensions, and valuations. Serre's exposition includes detailed treatments of complete discrete valuation fields, perfect residue fields, and higher-dimensional local fields. Each chapter builds systematically on previous material while maintaining precise mathematical rigor throughout. The work includes exercises and examples to reinforce key concepts. This influential text exemplifies Serre's characteristic style of presenting complex mathematical ideas with clarity and economy of expression. The book remains a foundational reference in the study of local fields and continues to influence modern research in arithmetic geometry.

Readers note this book provides a thorough treatment of local fields but requires significant mathematical maturity. The terse style and density of content make it challenging for self-study. Liked: - Clear progression from basic concepts to advanced topics - Precise mathematical exposition and proofs - Comprehensive coverage of local field theory fundamentals Disliked: - Limited examples and exercises - Assumes extensive background knowledge - Too concise for beginners - No motivation or context for theorems From Goodreads (3 ratings, 4.67/5 average): "Beautiful but difficult text. Not for first exposure to the subject." - Mathematics PhD student From Amazon.fr (2 ratings, 5/5 average): "Un classique pour comprendre la théorie des corps locaux, mais demande des bases solides en algèbre." (A classic for understanding local field theory, but requires solid algebra foundations.) The book appears infrequently on review sites due to its specialized advanced mathematics focus.

Local Fields by J.W.S. Cassels This text provides a systematic development of local field theory with applications to class field theory and explicit formulas.

Introduction to Cyclotomic Fields by Lawrence C. Washington The book presents the theory of cyclotomic fields and their connection to class field theory, p-adic L-functions, and Iwasawa theory.

Class Field Theory by Emil Artin These lecture notes establish the foundations of class field theory through a cohomological approach that complements Serre's perspective.

Basic Number Theory by André Weil This work develops local fields and adeles as part of a broader treatment of algebraic number theory and its foundations.

Algebraic Number Theory by Jürgen Neukirch The text builds local and global field theory from the ground up with connections to Galois theory and class field theory.

🔹 Corps Locaux (Local Fields) was published in 1962 and originated from Serre's lectures at Collège de France during 1958-1959, becoming a foundational text in algebraic number theory. 🔹 Jean-Pierre Serre, the author, became the youngest Fields Medal winner in history when he received it in 1954 at age 27, and remains the youngest recipient to this day. 🔹 The book introduces and develops the theory of local fields, which are crucial in modern number theory and have applications in cryptography and coding theory. 🔹 Despite being relatively slim at around 200 pages, Corps Locaux has been translated into multiple languages and remains a standard reference more than 60 years after its publication. 🔹 Serre wrote the book in a characteristically concise style that influenced how mathematical texts would be written for decades to come, emphasizing elegance and precision over exhaustive detail.