Algebraic Theory of Numbers and Representations by D. K. Faddeev and his co-authors represent a fascinating and deep dive into one of the most exciting areas of mathematics. This work will be a real discovery for students, graduate students, teachers and anyone interested in theoretical mathematics and its applications. This book explores the basic concepts of algebraic number theory, as well as their relationship to group representations. The authors offer the reader not only theoretical foundations, but also a lot of examples, tasks and exercises that will help to consolidate the knowledge gained. Each chapter is written with clarity and attention to detail, making complex ideas understandable. The book covers a wide range of topics, ranging from prime numbers to more complex structures such as number classes and algebraic numbers. The authors pay special attention to the representations of groups, which allows the reader to see how these concepts intertwine and interact with each other. This makes the material not only theoretically significant, but also practically useful for solving real problems. "Algebraic theory of numbers and representations" will be interesting both for those who are just beginning to get acquainted with mathematics, and for experienced researchers seeking to deepen their knowledge. Students of mathematical faculties will find in the book useful materials for preparing for exams, and teachers will be able to use it as a basic or additional teaching aid . In addition, lovers of mathematics, seeking to expand their horizons, will be able to enjoy deep reflections of the authors about the nature of numbers and their properties. The topics raised in the book go far beyond ordinary algebra. The authors explore the relationships between different areas of mathematics, allowing the reader to see how algebraic number theory can be applied in other disciplines, such as graph theory and combinatorics. This makes the book especially valuable for those who want to understand how different mathematical concepts interact with each other. Style D. K. Faddeev and his co-authors are distinguished by the clarity and logic of the presentation .. They are able to explain complex ideas in simple language, which makes their work available to a wide audience. Faddeev, known for his research in algebra and number theory, is also the author of other significant works, such as