The Fourier Transform and Equations over Finite Abelian Groups An introduction to the method of trigonometric sums LECTURE NOTES BY L´aszl´oBabai Department of Computer Science University of Chicago December 1989 Updated June 2002 VERSION 1.3 The aim of these notes is to present an easily accessible introduction to a powerful method of number theory. The punchline will be the following finite counterpart of Fermat's Last The- orem: Theorem 0.1 If k is an integer, q a prime power, and q k4 + 4, then the Fermat equation ≥ xk + yk = zk (1) has a nontrivial solution in the finite field Fq of order q. This result seems to belong to algebraic geometry over finite fields: we have an algebraic variety and we assert that it has points over Fq other than certain \trivial" ones. In fact, we can asymptotically estimate the number of solutions if q=k4 is large. As we shall see, algebraic equations have little to do with the method. In- deed, a much more general result will follow easily from the basic theory. Let F× = Fq 0 . q nf g Theorem 0.2 Let k be an integer, A1;A2 Fq; li = (q 1)= Ai (not neces- sarily integers), and assume that ⊆ − j j 2 q k l1l2 + 4: (2) ≥ Then the equation k x + y = z (x A1; y A2; z F×) (3) 2 2 2 q has at least one solution. k Theorem 0.1 follows from this result if we set A1 = A2 = a : a Fq×. q 1 f 2 Clearly, Ai = g:c:d:(−k;q 1) (q 1)=k and therefore li k in this case. j j − ≥ − ≤ Note that in Theorem 0.2, the sets A1 and A2 are arbitrary (as long as they are not too small compared to q). This result has a flavor of combinatorics where the task often is to create order out of nothing (i.e., without prior structural assumptions). Results like this one have wide applicability in combinatorial terrain such as combinatorial number theory (to which they belong) and even in the theory of computing. 1 Notation C: field of complex numbers C× = C 0 : multiplicative group of complex numbers nf g Z: ring of integers Zn = Z=nZ: ring of mod n residue classes Fq: field of q elements where q is a prime power (Fq; +): the additive group of Fq F× = Fq 0 : the multiplicative group of Fq. q nf g 1 Characters Let G be a finite abelian group of order n, written additively. A character of G is a homomorphism χ : G C× of G to the multiplicative group of (nonzero) complex numbers: ! χ(a + b) = χ(a)χ(b)(a; b G): (4) 2 Clearly, χ(a)n = χ(na) = χ(0) = 1 (a G); (5) 2 so the values of χ are nth roots of unity. In particular, 1 χ( a) = χ(a)− = χ(a) (6) − where the bar indicates complex conjugation. The principal character is defined by χ0(a) = 1 (a G): (7) 2 Proposition 1.1 For any nonprincipal character χ of G, χ(a) = 0: (8) X a G 2 Proof: Let b G be such that χ(b) = 1, and let S denote the sum on the left hand side of equation2 (8). Then 6 χ(b) S = χ(b)χ(a) = χ(b + a) = S · X X a G a G 2 2 hence S(χ(b) 1) = 0; − proving the claim. 2 Corollary 1.2 (First orthogonality relation for characters) Let χ and be two characters of G. Then n if χ = χ(a) (a) = (9) X 0 otherwise. a G 2 Proof: The case χ = follows from equation (6). If χ = , then χψ is a 6 nonprincipal character, hence Proposition 1.1 applies. As observed in the last proof, the pointwise product of the characters χ and is a character again: (χψ)(a) := χ(a) (a) (10) Let G denote the set of characters. It is easy to see that this set forms an abelian groupb under operation (10). G is called the dual group of G. b th Proposition 1.3 Let ! be a primitive n root of unity. Then the map χj : Zn C× defined by ! ja χj(a) := ! (11) is a character of Zn for every j Z. Moreover, 2 (a) χj = χk if and only if j k mod n; j ≡ (b) χj = χ1; (c) Zn = χ0; : : : ; χn 1 . f − g b Z Z (d) Consequently, n ∼= n. b Proof: (a) and (b) are straightforward. Let now χ be an arbitrary character; j then χ(1) = ! for some j; 0 j n 1 by eqn. (5). If follows that χ = χj. ≤ ≤ − Now, (d) is immediate. Proposition 1.4 If G is a direct sum: G = H1 H2, and 'i : Hi C× is a ⊕ ! character of Hi (i = 1; 2), then χ = '1 '2, defined by ⊕ χ(h1; h2) := '1(h1) '2(h2); (12) · is a character of G. Moreover, all characters of G are of this form. Conse- quently, G = H1 H2 (13) ∼ ⊕ b b b Proof: The first statement is clear, and it is easy to verify that the map H1 H2 G defined by (12) is injective. Let now χ G. The restriction ⊕ ! 2 'bi = χbH is clearlyb a character of Hi, and it is easy to verifyb that χ = '1 '2. j i ⊕ Corollary 1.5 G ∼= G. b 3 Proof: G = Zn Zn , hence G = Zn Zn = G using the previous ∼ 1 ⊕· · ·⊕ k ∼ 1 ⊕· · ·⊕ k ∼ two propositions. b b b We remark that there is no natural isomorphism between G and G; even for cyclic groups, the isomorphism selected depends on the arbitrary choiceb of !. The consequent isomorphism G ∼= G is, however, natural: b Corollary 1.6 G can be identified with G in the following natural way: for b a G, define a~ G by b 2 2 b a~(χ) = χ(a)(χ G): (14) 2 b The map a a~ is an isomorphism of G and G. 7! b Proof: Left to the reader. Let CG denote the space of functions f : G C. This is an n-dimensional linear space over C. We introduce an inner product! over this space: 1 (f; g) = f(a)g(a)(f; g CG): (15) n X 2 a G 2 Theorem 1.7 G forms an orthonormal basis in CG. b Proof: Orthonormality follows from Cor. 1.2. Completeness follows from Cor. 1.5 which implies that G = n = dim(CG). j j Let χ0; : : : ; χn 1 be the charactersb of G = a0; : : : ; an 1 . The n n matrix − f − g × C = (χi(aj)) (16) is the character table of G. 1 Corollary 1.8 The matrix A = pn C is unitary, i. e., AA∗ = A∗A = I.(A∗ is the conjugate transpose of A; I is the n n identity matrix.) × Proof: A∗A = I follows immediately from Theorem 1.7 in view of the for- mula (15). Corollary 1.9 (Second orthogonality relation for characters) Let a; b G. Then 2 n if a = b χ(a)χ(b) = (17) X 0 otherwise. χ G 2 b First proof: This is a restatement of the fact that AA∗ = I in Corollary 1.8. Second proof: In view of the identification of G and G (Cor. 1.6), Cor. 1.9 is b a restatement of Cor. 1.2 for the abelian group G in placeb of G. We state a special case separately. The followingb is the dual of Proposi- tion 1.1. 4 Corollary 1.10 For any non-zero element a G, 2 χ(a) = 0: X χ G 2 b 2 Fourier Transform Corollary 2.1 Any function f CG can be written as a linear combination of characters: 2 f = c χ. (18) X χ χ G 2 b Such a linear combination is also called a trigonometric sum since f(a) is th expressed as a combination of n roots of unity. The coefficients cχ are called the Fourier coefficients and are given by the formula cχ = (χ, f): (19) Proof: Expansion (18) exists by Theorem 1.7. The inner product (χ, f) is equal to cχ by orthonormality. The function f : G C, defined by ! b b f(χ) = ncχ = χ(a)f(a)(χ G); (20) X 2 b a G b 2 is called the Fourier Transform of f. This transformation is easily inverted: using equations (18) and (20), we see that 1 f = c χ = f(χ)χ, X χ X n χ G χ G b 2 2 b b hence the formula for the Inverse Fourier Transform is 1 f(a) = f(χ)χ( a)(a G): (21) n X − 2 χ G b 2 b We derive a simple consequence. Let δ CG be defined by 2 1 if a = 0 δ(a) = (a G): 0 if a = 0 2 6 Corollary 2.2 (a) δ(χ) = 1 (χ G): (22) 2 b b 5 (b) 1 δ = χ. (23) n X χ G 2 b Proof: (a) follows from eqn. (20). (b) follows from eqn. (21). (Note that (b) also follows from the second orthogonality relation (17) with a = 0.) Applying formula (15) to G we obtain the inner product 1b (f; g) = f(χ)g(χ)(f; g CG) (24) n X 2 b χ G 2 b over the space CG. Corollary 1.8 tells us that Fourier transformation is pn b times a unitary transformation between CG and CG: b Theorem 2.3 (Plancherel formula) For any f; g CG, 2 (f; g) = n(f; g): (25) b b First proof: Using the notation introduced before Cor.