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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.1 Algebraic Numbers

Barry Mazur

The roots of our subject go back to ancient Greece while its branches touch almost all aspects of contemporary mathematics. In 1801 the Disquisitiones Arithmeticae Of CARL FRIEDRICH GAUSS [VI.26] was first published, a “founding treatise,” if ever there was one, for the modern attitude toward number theory. Many of the still unachieved aims of current research can be seen, at least in embryonic form, as arising from Gauss’s work.

This article is meant to serve as a companion to the reader who might be interested in learning, and thinking about, some of the classical theory of algebraic numbers. Much can be understood, and much of the beauty of algebraic numbers can be appreciated, with a minimum of theoretical ...

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