Quadratic Form Gauss Sums
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Let p be a prime, n, r positive integers, S an integer coprime to p. We let Q_r denote an r-dimensional integral quadratic form. For convenience, set e(x) = e^{2 pi i x}, where x is any rational number, i is the imaginary unit. Denote the quadratic Gauss sum by G(S;p^n). The evaluation of this sum was completed by Gauss in the early 19th century. Many proofs of these results have subsequently been obtained through a variety of methods. We are interested in the so called quadratic form Gauss sum, given by G(Q_r;S;p^n) - \sum_{x_1, x_2, ..., x_r}^{p^n-1} e(S/p^n * Q_r). Under certain assumptions on Q_r, we show how we may express G(Q_r;S;p^n) as a product of quadratic Gauss sums.
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Copyright © 2016 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, research, scholarship, and teaching. Theses may only be shared by linking to Carleton University Institutional Repository and no part may be used without proper attribution to the author. No part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner.
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