Number theory is an old and difficult subject. It is possible to have a problem that is easy to formulate (e.g. Fermat’s last theorem) but very difficult to solve. A variety of different methods have been developed over the centuries. This book is devoted to methods in number theory connected with (hidden) symmetries, their realizations by means of groups and their representations. The book is designed for a wide audience of non-specialists. The authors were willing to make an attempt to explain exciting discoveries in mathematics for a larger public without special training in mathematics. They did a splendid job. They are able to describe an eminent role of symmetries in number theory by carefully explaining what it means to represent something (a group) by something else. The first part of the book reviews basic algebraic notions and introduces the Legendre symbol and the law of quadratic reciprocity. The Galois group and its role in number theory is the main topic of the second part. The last part treats various reciprocity laws. It also indicates the prominent role played by ideas of symmetry in the proof of Fermat’s last theorem. The book describes very nicely and in simple terms key ideas of the field so that they can be appreciated by people with no particular mathematical education. It is also inspiring and useful for general mathematicians who are not specialized in the field.

Reviewer:

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