By Igor E. Shparlinski
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May 7, 2002 23:25 WSPC/Guidelines ExpSums-Intro Igor E. 9: Show that for any arbitrary subsets X , Y ∈ IFp and any nontrivial multiplicative character χ, 1/2 χ(x + y) ≤ (p#X #Y) . 10: Show that for any nontrivial multiplicative character χ and a ≡ 0 (mod p) p χ(x)ep (ax) = p1/2 . x=0 Hint: For any y ≡ 0 (mod p), p p χ(x)ep (ax) = x=0 χ(xy)ep (ayx). x=0 therefore 2 p p−1 χ(x)ep (ax) (p − 1) 2 p χ(x)ep (bx) . 11: Let n|p − 1 and Ωn be the set of all multiplicative characters χ for which χn is the trivial character, χn = χ0 .
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