Rings of Integers, Gauss-jacobi Sums, and Their Applications
نویسنده
چکیده
In this paper we shall explore the structure of the ring of algebraic integers in any quadratic extension of the field of rational numbers Q, develop the concepts of Gauss and Jacobi sums, and apply the theory of algebraic integers and that of Gauss-Jacobi sums to solving problems involving power congruences and power sums as well as to proving the quadratic and cubic reciprocity laws. In particular, we shall address the problem of when a rational prime (that is, a prime in Z) stays a prime in the ring of algebraic integers in any quadratic extension of Q, discuss when a rational prime can be written as the sum of two squares, and find the number of solutions to congruence equations of the form xn + yn ≡ 1 mod p when n = 2 or 3.
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تاریخ انتشار 2012