Quadratic residues and Gauss sums
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Quadratic residues and Gauss sums book
This chapter is centered on quadratic reciprocity and the theory around it. In Section 3.1, we start with an elementary theory of power residues based on grouptheoretic arguments. Then we specialize to quadratic residues. In Section 3.3, we prove Gauss’ quadratic reciprocity law and (again) Fermat’s theorem on sums of two squares. In our presentation, we disregard the historical development of the subject. For a readable appreciation of the historical development we refer the interested reader to the books  and . In this volume, we use Gauss sums to give a concise proof of the quadratic reciprocity law, and we use Jacobi sums for a proof of the theorem on sums of two squares (compare to the presentations given in ,  and, even more detailed, in ). Gauss and Jacobi sums are introduced in Section 3.2, and a thorough presentation of the theory of quadratic characters is given in Section 3.5. All this material however does not only permit a simple proof of the quadratic reciprocity law, but has wide inﬂuence on the development of the following chapters. It will be used for the theory of Dirichlet series in Chapter 4, for Gauss’ genus theory of binary quadratic forms in Chapter 6, and for the theory of biquadratic residues in Chapter 7.