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11R11 Number theory -- Algebraic number theory: global fields -- Quadratic extensions

11A55 Number theory -- Elementary number theory -- Continued fractions

11E16 Number theory -- Forms and linear algebraic groups -- General binary quadratic forms

11Rxx Number theory -- Algebraic number theory: global fields En-ligne: Springerlink | Zentralblatt | MathSciNet

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his book presents a smooth introduction to the first principles of algebraic number theory. After a review of elementary number theory, a few quadratic number rings are presented as examples of Euclidean rings and rings without unique factorization. Chapter 2 is a “crash course in ring theory” and introduces the readers to ideals; Chapter 3 is a geometric counterpart and deals with the theory of lattices. In Chapter 4, the rings of integers of quadratic number fields are studied, and it is proved that the nonzero ideals in these rings are Dedekind domains, e.g. admit unique factorization into prime ideals. Next in line is the Minkowski’s theorem on the finiteness of the class group. Chapter 6 introduces continued fractions and explains the connection with units in real quadratic number fields, and the last chapter is a brief introduction to Bhargava’s presentation of Gauss’s theory of composition of primitive binary quadratic forms.

The book is very well written, has worked examples and numerous exercises, and should easily accessible to readers with a good background in elementary number theory and the most basic algebraic notions. I highly recommend it to everyone interested in number theory beyond the most basic level. (Zentralblatt)

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