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Sign Ambiguities of Gaussian Sums Heon Kim Louisiana State University and Agricultural and Mechanical College, Hkim@Math.Lsu.Edu Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2007 Sign Ambiguities of Gaussian Sums Heon Kim Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Applied Mathematics Commons Recommended Citation Kim, Heon, "Sign Ambiguities of Gaussian Sums" (2007). LSU Doctoral Dissertations. 633. https://digitalcommons.lsu.edu/gradschool_dissertations/633 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. SIGN AMBIGUITIES OF GAUSSIAN SUMS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Heon Kim B.S. in Math., Chonbuk National University, 1995 M.S. in Math., Chonbuk National University, 1997 M.A. in Math., University of Georgia, 2002 December 2007 Acknowledgments This dissertation would not be possible without several contributions. The love of family and friends provided my inspiration and was my driving force. It has been a long journey and completing this work is definitely a high point in my academic career. I could not have come this far without the assistance of many individuals and I want to express my deepest appreciation to them. It is a pleasure to thank my dissertation advisor, Dr. Helena Verrill, and coad- visor Dr. Paul van Wamelen for their guidance and suggestions throughout the period of this research. I could not finish without their warm concern and contin- uous encouragement. I really appreciate them. Thank you very much. I wish to thank my dissertation committee members, Dr. Robert Perlis, Dr. Richard Litherland, Dr. Padmanabhan Sundar, Dr. Charles Delzell, and Dr. Martin Feldman for their suggestions. It is a pleasure also to thank Dr. Leonard Richard- son, Dr. James Oxley, Dr. Lawrence Smolinsky, Dr. George Cochran, Dr. Guillermo Ferreyra for providing me with a pleasant working environment. A special thanks to my Korean colleagues and friends Dr. Hwashin Park, Dr. Jongjin Kim, Dr. Jaeup So, Dr. Kyungshik Seo, Dr. Daeyeul Kim, Dr. Shik Lee, Dr. Jongjin Park, Dr. Jiyeun Won, Dr. Daewoong Lee, Dr. Jeongbae Lee, Dr. He- unggi Park, Dr. Jongsook Keum, Dr. Eunhee Lee, Dr. Jeongha An, Dr. Soomi Jung, Dr. Daekyu Kim, Dr. Sanghyun Kim, Dr. Daeshik Park, Dr. Jeonghoon Park, Dr. Sangyeup Lee, Dr. Iwasa Akira, Dr. Youngwon He, Pastor. Kitae Nam, Dr. Sungchul No, Dr. Jongho Uhm, Dr. Jongseob Rim, Dr. Jeonghan Kimn, Dr. Jinhoon Lee, Dr. Wonjoon Jung, Dr. Hakchul Shin, Dr. Gangsun Lee, Jaeyoung Kim, Taehee Han, Seokwon Kim, Hoonshin Jung, Youngjeon Cho, Hyoshik Kim, ii Hongshik Kim, Kwansoo Kim, Woonsoo Lee, Changdeuk Seo, Jeongho Choi, Joon- pyo Hong, Kyeungsoo Han, Changwon Oh, Jaeyoung Kwak for their warm concern and encouragement. This dissertation is dedicated to my whole family members. My parents, Moon- hong Kim and Jeongok Eun, have always believed me and helped me reach my goals. Their support forged my desire to achieve all that I could in life. I owe them everything and wish I could show them just how much I love and appreciate them. My wife, Hyeyoung Ko, whose love and encouragement allowed me to finish this journey, already has my heart so I will just give her a heartfelt “thanks.” My lovely daughter and son, Sol (Carol) and San (Kevin), have always gave me smile, pride, and happiness. I really love you guys and always proud of you. My parant-in-law, Seokdo Ko and Soonrae Kang, always gave me encouragement and faithfulness. Thanks for your patience. I also deeply appreciate to my brothers, Hoon Kim (Soojin Yoon, Minji, Jooun), Hyeok Kim (Joohoon Park, Yoonji, Gun), and sister-in-law, Eunjoo Ko (Yongkyu Choi, Sunhwa, Yeni), and brothers-in-law, Hojoon Ko (Sunmi Choi), Bumjin Ko (Insook Choi, Gun, Seyoung) for their end- less belief and love. Finally, I would like to dedicate this work to my lost “father,” Moonhong Kim, who left us too soon. I hope that this work makes you proud. iii Table of Contents Acknowledgments . ii Abstract . v Introduction ..................................................... 1 1. Preliminaries . 5 1.1 Quadratic Reciprocity . 5 1.2 Extensions of Prime Ideals . 7 2. Gauss Sums over Fp ............................................ 10 2.1 Cyclotomic Fields . 10 2.2 Stickelberger’s Congruence for Gauss Sums . 12 3. Multiplicative Relations . 14 3.1 Multiplicative Relations of Gaussian Sums . 14 3.2 Computational Algorithm . 16 4. Yamamoto’s Sign Ambiguities . 22 4.1 Half Sets . 22 4.2 Some Lemmas . 28 4.3 Yamamoto’s Sign Ambiguity . 39 5. Main Result . 43 5.1 Explicit Signs . 43 5.2 Sign Ambiguities . 47 References . 55 Vita ............................................................. 56 iv Abstract In 1934, two kinds of multiplicative relations, norm and Davenport-Hasse relations, between Gaussian sums, were known. In 1964, H. Hasse conjectured that the norm and Davenport-Hasse relations are the only multiplicative relations connecting the Gaussian sums over Fp. However, in 1966, K. Yamamoto provided a simple counterexample disproving the conjecture when Gaussian sums are considered as numbers. This counterexample was a new type of multiplicative relation, called a sign ambiguity (see Definition 3.3), involving a ± sign not connected to elementary properties of Gauss sums. In Chapter 5, we provide an explicit product formula giving an infinite class of new sign ambiguities and we resolve the ambiguous sign by using the Stickelberger’s theorem. v Introduction In mathematics, the general theory of Gauss sums was developed in the early nineteenth century, with the use of Jacobi sums and their prime decomposition in cyclotomic fields. A Gaussian sum is a particular kind of finite sum of roots of unity. The case originally considered by Carl Friedrich Gauss, in his Disquisitions Arithmeticae 1801, was the quadratic Gauss sums. The quadratic Gauss sums are certain sums over exponential functions with quadratic argument. They are named after C. F. Gauss, who studied them extensively. Definition 0.1. The quadratic Gauss sum G(m, k) over the ring of integers modulo k is defined by k−1 X G(m, k) = exp(2πimn2/k), n=0 where exp(x) is the exponential function. This sum is hard to evaluate, even in the special case that m = 1 and k is an odd positive integer. But, in this case, Gauss showed that this sum has the value √ √ ± k if k ≡ 1 mod 4 and ±i k if k ≡ 3 mod 4. The determination of the sign is a much more difficult problem. The conjecture that the plus sign holds in each case was made by Gauss and recorded in his diary in May 1801. It was not until four years later that he found a proof. Subsequently, he was able to evaluate his quadratic sum for all positive integers k. Theorem 0.2. (The evaluation of the quadratic Gauss sums) If m = 1 and k is an odd positive integer, then √ k if k ≡ 1 mod 4 G(1, k) = √ i k if k ≡ 3 mod 4. 1 Let e be a positive integer, e > 2, p be a prime number such that p ≡ 1 mod e th and ζe be a primitive e root of unity. Let Fp be the finite field of p elements. Then we identify Fp with the residues 0, 1, 2, ··· , p − 1 mod p. Denote Fp − {0} ∗ by Fp; that is a cyclic multiplicative group of order p − 1. The following definition generalizes the quadratic Gauss sum. Definition 0.3. Let χ be a multiplicative character (see Definition 2.6) on Fp. For a ∈ Z, the Gauss sum τ(a) is defined by X a t τ(a) = χ (t)ζp. t∈Fp Many applications of Gauss sum evaluations are investigated; not only to number theory but also to physics and to such areas of mathematics as graph theory and combinatorics, operator theory, coding theory, cryptography, combinatorial designs, analysis and algebra. One powerful approach to simplifying the evaluation of Gauss sums is to study multiplicative relations, defined in Section 3.1, between them. The most basic mul- tiplicative relations are the norm relation and Davenport-Hasse product formula, say D-H relation, for Gauss sums. In [3, p.465], H. Hasse conjectured that all the multiplicative relationships of Gaussian sums over Fp can be deduced from the norm and D-H relations. However, in [14], K. Yamamoto provided a simple counterexample disproving the conjecture if we consider Gauss sum, as numbers rather than ideals. This counterexample was a new type of multiplicatively independent relationship of Gaussian sums involving an ambiguous sign not direct consequences of the norm and D-H relations. In [15], K. Yamamoto succeeded in not only proving that in- finitely many exist, but also produced a formula giving the exact “number” of sign ambiguities to be expected in each composite number e (≥ 3) (see Theorem 2 3.4). Shortly thereafter, working in the context of Jacobi sums and with the aid of a computer, further sign ambiguities were discovered by Muskat, Whiteman, and Zee (see [8, 9, 10]) when e = 15, 20, 21, 24, 28, 39, 55, 56. In 2002, B. Murray (see [7, Theorem 3.54]) provided an infinite class of sign ambiguities when e = q1q2, where q1 and q2 are prime numbers such that q1 ≡ 5 mod 8, q2 ≡ 3 mod 4 and q2 is a biquadratic residue modulo q1. In [15, § 5], K. Yamamoto said that the case 2|e, 4 - e is reducible to the case where e is replaced by an odd number e/2.
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