fourier - The Fourier Transform and Equations over Finite...

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Unformatted text preview: The Fourier Transform and Equations over Finite Abelian Groups An introduction to the method of trigonometric sums LECTURE NOTES BY L´ aszl´o Babai 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 ≥ k 4 + 4 , then the Fermat equation x k + y k = z k (1) has a nontrivial solution in the finite field F q 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 F q other than certain “trivial” ones. In fact, we can asymptotically estimate the number of solutions if q/k 4 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 × q = F q \{ } . Theorem 0.2 Let k be an integer, A 1 ,A 2 ⊆ F q ,l i = ( q- 1) / | A i | (not neces- sarily integers), and assume that q ≥ k 2 l 1 l 2 + 4 . (2) Then the equation x + y = z k ( x ∈ A 1 ,y ∈ A 2 ,z ∈ F × q ) (3) has at least one solution. Theorem 0.1 follows from this result if we set A 1 = A 2 = { a k : a ∈ F × q . Clearly, | A i | = q- 1 g . c . d . ( k,q- 1) ≥ ( q- 1) /k and therefore l i ≤ k in this case. Note that in Theorem 0.2, the sets A 1 and A 2 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 \{ } : multiplicative group of complex numbers Z : ring of integers Z n = Z /n Z : ring of mod n residue classes F q : field of q elements where q is a prime power ( F q , +): the additive group of F q F × q = F q \{ } : the multiplicative group of F q . 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) Clearly, χ ( a ) n = χ ( na ) = χ (0) = 1 ( a ∈ G ) , (5) so the values of χ are n th roots of unity. In particular, χ (- a ) = χ ( a )- 1 = χ ( a ) (6) where the bar indicates complex conjugation. The principal character is defined by χ ( a ) = 1 ( a ∈ G ) . (7) Proposition 1.1 For any nonprincipal character χ of G , X a ∈ G χ ( a ) = 0 . (8) Proof: Let b ∈ G be such that χ ( b ) 6 = 1, and let S denote the sum on the left hand side of equation (8). Then χ ( b ) · S = X a ∈ G χ ( b ) χ ( a ) = X a ∈ G χ ( b...
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