Gauss and Jacobi Sums
In Chapter 6 we introduced the notion of a quadratic Gauss sum. In this chapter a more general notion of Gauss sum will be introduced. These sums have many applications. They will be used in Chapter 9 as a tool in the proofs of the laws of cubic and biquadratic reciprocity. Here we shall consider the problem of counting the number of solutions of equations with coefficients in a finite field. In this connection, the notion of a Jacobi sum arises in a natural way. Jacobi sums are interesting in their own right, and we shall develop some of their properties.
To keep matters as simple as possible, we shall confine our attention to the finite field ℤ/pℤ = F p and come back later to the question of associating Gauss sums with an arbitrary finite field.
KeywordsFinite Field Unique Factorization Multiplicative Character Legendre Symbol Nonzero Complex Number
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