Algebraic Theory of Numbers

Algebraic Theory of Numbers presents the foundational concepts of algebraic number theory, beginning with the basic properties of integers and progressing through increasingly complex number fields. The text covers prime factorization, ideals, and the arithmetic of algebraic integers. Samuel develops the material through a series of exercises integrated with the theoretical content, allowing readers to engage directly with key concepts and proofs. The book maintains a focused scope, concentrating on essential topics while avoiding unnecessary abstraction or generalization. The presentation follows a natural progression from elementary number theory through to more advanced topics like Dedekind domains and ramification theory. Each chapter builds systematically on previous material while introducing new tools and techniques. This concise volume serves as both an introduction to algebraic number theory and a bridge to more advanced studies in the field. The work emphasizes connections between abstract concepts and concrete examples, highlighting the interplay between basic number theory and its algebraic extensions.

Readers describe this as a concise introduction to algebraic number theory that requires strong prerequisites in abstract algebra. Several reviews note it works best as a supplementary text rather than primary learning source. Liked: - Clear explanations of core concepts - Efficient presentation in under 200 pages - Quality exercises with solutions - Logical progression of topics Disliked: - Too terse for self-study - Assumes significant background knowledge - Some proofs lack detail - Dated notation (published 1970) One reader on Amazon noted: "The book moves quickly and leaves many details to the reader - this is both a strength and weakness depending on your level." Ratings: Goodreads: 4.0/5 (22 ratings) Amazon: 4.3/5 (15 ratings) Multiple reviewers recommend pairing it with more comprehensive texts like Marcus' "Number Fields" for a complete understanding of the material.

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A Classical Introduction to Modern Number Theory by Kenneth Ireland, Michael Rosen The book builds from elementary number theory to advanced concepts in algebraic number theory and links to other mathematical areas.

Number Fields by Daniel A. Marcus This work presents the fundamentals of algebraic number theory with emphasis on unique factorization, ideals, and ring theory.

Algebraic Number Theory by Serge Lang The text develops the foundations of algebraic number theory from first principles through class field theory with detailed proofs.

A Course in Arithmetic by Jean-Pierre Serre This concise treatment connects elementary number theory to advanced concepts in quadratic forms and local fields.

🔢 Pierre Samuel (1921-2009) was a prominent member of the Bourbaki group, an influential collective of mathematicians who revolutionized the way mathematics was presented and taught in the 20th century. 📚 The book, first published in 1967, emerged from Samuel's lectures at University of Clermont-Ferrand and became a standard reference for introducing students to algebraic number theory. 🎓 Despite its relatively slim size (less than 200 pages), this book covers fundamental concepts that typically require much lengthier treatments in other texts, making it particularly valued for its conciseness. 🌍 The text has been translated into multiple languages and remains in print after more than 50 years, testament to its enduring pedagogical value in mathematics education. 💡 The book's approach to algebraic number theory helped bridge the gap between abstract algebra and number theory, influencing how these subjects would be taught in universities worldwide.