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Notes on Character Sums NOTES ON CHARACTER SUMS WEN WANG Abstract. In this article, the properties of character sums, including Gauss sums and Jacobi sums are investigated. 1. Introduction{Historical Notes The origin of the Gauss sum and Jacobi sum in the work of C.F. Gauss and C.G.J. Jacobi. Gauss introduced the Gauss sum in his Disquisitione Arithmeticae[Ga1] in July, 1801, and Jacobi introduced the Jacobi sum in a letter to Gauss[Ja1] dated February 8, 1827. The sum introduced by Gauss in 1801 is kX−1 e2휋imn2=k; n=0 which is now called a quadratic Gauss sum. This sum is not easy to evaluate, even in the special case that m = 1 and k is an odd positive integer.p Inp this case, Gauss was easily able to show that this sum has the value ± k or ±i k, according as k is of the form 4u + 1 or 4u + 3, respectively. Specific examples convinced Gauss that the plus sign is always correct. On August 30, 1805, Gauss wrote in his diary he was able to prove his conjecture on the sign of these sums. A few years later, Gauss[Ga2] published an evaluation of his quadratic Gauss sum for all positive integer k. In his study of primes in arithmetic progressions, G.L. Dirichlet [Di1] introduced the multiplicative character 휒 modulo k and the sum kX−1 G(휒) = e2휋imn=k: n=0 This is also called a Gauss sum, as it coincides with the quadratic Gauss sum above in the case that 휒 has order 2 and k is a prime not dividing m. The sum now called a Jacobi sum, which is in essence the one Jacobi introduced in 1827, is Date: 2010-03-12. Key words and phrases. Character sums, Finite fields. The calculation in section 9 of this article is done by SAGE, an open source mathematical calculation software program directed by William Stein. See http://www.sagemath.org/. 1 2 WEN WANG X J(휒, ) = 휒(n) (1 − n); n mod p where 휒 and are multiplicative characters modulo a prime p. Jacobi was already aware in 1827 that Gauss and Jacobi sums are related like gamma and beta functions, i.e., for k = p and m = 1. J(휒, ) = G(휒)G( )=G(휒휓) when 휒휓 is non-trivial. After the initial work of Gauss, Dirichlet, and Jacobi, many well-known math- ematicians made contributions to the theory of Gauss and Jacobi sums, including L. Carlitz, A. Cauchy, S. Chowla, H. Davenport, G. Eisenstein, B. Gross, H. Hasse, N.M. Katz, N. Koblitz, L. Kronecker, E.E. Kummer, D.H. and E. Lehmer, I.J. Mordell, S.J. Patterson, C.L. Siegel, L. Stickelbegger, and A. Weil. In the following sections, Gauss and Jacobi sums are introduced via the language of characters over finite fields. 2. Characters of finite fields Suppose F be a finite field with q = pd elements, where p is a prime number. It is obvious that Fp = Z=Zp and therefore F is a dimension d vector space over ∗ Fp. We say that 휒 is a character of F , or just of F if there is no ambiguity, if 휒 is a multiplicative group homomorphism of F into C∗. 휒 is called primitive if it is injective. Note that F ∗ is a cyclic group with a generater, say, g. Then 휒 is determined by 휒(g) 2 C∗. If 휒 is primitive, one may choose g such that 2휋i ∗ ∗ 휒(g) = e p−1 . Note that Hom(F ; C ) has a natural abelian group structure, i.e., for 휒, 2 Hom(F ∗; C∗), 휒휓(a) = 휒(a) (a), for any a 2 F ∗. It is not hard to verify that the identity for Hom(F ∗; C∗) is the character that maps every element of F ∗ to 1 2 C, we shall call this character the identity character and denote it by 1. For a given character, let m be the smallest positive integer such that 휒m = 1. Hence m divides q − 1. It follows that there is a primitive order m character of F if and only if m divides q − 1. For example, if F is a prime field and charF > 2, then 2 j p − 1, therefore there exist an order 2 character of F . Since F ∗ is cyclic, therefore( this) order 2 character is unique, which is the well-known the Legendre ⋅ symbol: p , which maps a quadratic residue to 1 and a non-residue to −1. We say that 휏 is an additive character if 휏 is a group homomorphism from the additive group of F into the multiplicative group C∗. Note that the trace map tr is an additive map, which will be used for the construction of the Gauss sum in the next section. MATH 414 COURSE PROJECT 3 3. Gauss Sums Let F be a finite field with q elements, and let tr be the trace map from F to the prime field Fp. The Gauss sum of a character 휒 of F is defined as follows: X trt G(휒) := 휒(t)휁p : t2F In general, according to an additive character 휏, we can form the Gauss sum in the following way: X Ga(휒; 휏) = 휒(t)휏(at) t2F for some fixed a 6= 0 2 F . Note that if we consider the case when F itself is the prime field Fp, then the Gauss sum becomes: X at Ga(휒) := 휒(t)휁p : t2F For now on and to the end of this section we will consider the case when F is the prime field. Note that 휒(t) is a power of 휁p−1 for any 휒 and any t 2 F , therefore the Gauss sum G(휒) 2 Q(휁p−1; 휁p). We also have the following basic property of the Gauss sums[Ir1, Co1]: p Proposition 3.1. If 휒 6= ", then jG(휒)j = p. Proof. The ideal is to evaluate the sum X Ga(휒)Ga(휒) a in two ways. −1 If a 6= 0, then Ga(휒) = 휒(a)G(휒) and Ga(휒) = 휒(a) G(휒). Thus Ga(휒)Ga(휒) = 2 2 jG(휒)j . Since G0 = 0, the sum evaluates as (p − 1)jG(휒)j . On the other hand, X X ax−ay Ga(휒)Ga(휒) = 휒(x)휒(y)휁 : x y Summing both sides over a yields, X X sumaGa(휒)Ga(휒) = 휒(x)휒(y)훿(x; y)p = (p − 1)p; x y where 훿(x; y) = 1 if x = y and zero otherwise. p Hence jG(휒)j = p. □ 4 WEN WANG 4. Jacobi Sums The Jacobi sum was first introduced by Jacobi in a letter to Gauss, which is ∗ defined as following. For two multiplicative character on Fp, we defined the Jacobi sum to be P J(휒, ) = t 휒(t) (1 − t) Here are some basic properties of the Jacobi sum: Proposition 4.1. Let " be the trivial character. Then, (a) J("; ") = p. (b) J("; 휒) = 0. (c) J(휒, 휒−1) = −휒(−1). (d) If 휒휓 6= ", then G(휒)G( ) J(휒, ) = : G(휒휓) Proof. Part (a) is immediate. Part (b) is a consequence of Lemma 7.3. To prove (c), notice that ( ) X X (a) X a J(휒, 휒−1) = 휒(a)휒−1(b) = 휒 = 휒 : b 1 − a a+b=1 a+b=1;b6=0 a6=1 Set a=(1 − a) = c. If c 6= 1, then a = c=(1 + c). It follows that as a varies over Fp, except the element 1, then c varies over Fp, except for −1. Thus X J(휒, 휒−1) = 휒(c) = −휒(−1): c6=−1 To prove (d), notice that ! ! X X G(휒)G( ) = 휒(x)휁x (y)휁(y) x y X = 휒(x) (y)휁x+y x;y ! X X = 휒(x) (y) 휁t: t x+y=t If t = 0, then X X X 휒(x) (y) = 휒(x) (−x) = (−1) 휒휓(x) = 0; x+y=0 x x since 휒휓 6= " by assumption. If t 6= 0, define x0 and y0 by x = tx0 and y = ty0. If x + y = t, then x0 + y0 = 1. It follows that MATH 414 COURSE PROJECT 5 X G(휒)G( ) = 휒휓(t)J(휒, )휁t = J(휒, )G(휒휓): t Substituting into Equation (7.1) yields G(휒)G( ) = J(휒, )G(휒휓): □ We have the following consequence p Corollary 4.2. If 휒, , 휒휓 are all not equal to ", then jJ(휒, )j = p. Proof. Clear from Proposition 4.1. □ 5. The Analogy between Gauss and Jacobi sums and Gamma and Beta functions Gross and Koblitz [Ko1, Ko2]noted that there is an analogy between the Gauss sums and Jacobi sums, to the Gamma and Beta functions respectively. In this section the analogy is stated briefly. Recall the (real) gamma function on x > 0 is defined as: Z 1 Γ(x) = tx−1e−tdt; 0 and the (real) beta function on x > 0; y > 0 is defined as: Z 1 B(x; y) = tx−1(1 − t)x−1dt: 0 Note that the relation between Beta and Gamma function could be written as: Γ(x)Γ(y) B(x; y) = Γ(x + y) R Here comes the analogy: the integration dt could be viewed as a sum on the domain (0; 1). Note that t 7! tx−1 is a multiplicative character from the multi- −t plicative group of R≥0 to itself, determined by x. Also note that e is an additive character from [0; +1) to [0; 1].
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