Quadratic Residues and Non-Residues: Selected Topics
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by
Steve Wright
2016
Abstract
Number theory as a coherent mathematical subject started with the work of
Fermat in the decade from 1630 to 1640, but modern number theory, that is, the
systematic and mathematically rigorous development of the subject from
fundamental properties of the integers, began in 1801 with the appearance of
the landmark text of Gauss, Disquisitiones Arithmeticae. A major part of the
Disquisitiones deals with quadratic residues and nonresidues. Beginning with
these fundamental contributions of Gauss, the study of quadratic residues and
nonresidues has subsequently led directly to many of the key ideas and
techniques that are used everywhere in number theory today, and the primary
goal of these lectures is to use this study as a window through which to view
the development of some of those ideas and techniques. In pursuit of that goal,
we will employ methods from elementary, analytic, and combinatorial number
theory, as well as methods from the theory of algebraic numbers.
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