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Question - Math 324 Assignment 8 (due November 12)

1. (6) If χ is a non-principal Dirichlet character mod q, show that ∑ a mod q χ(a) = 0.
Hint: What happens to ∑ a mod q χ(a) if it is multiplied by χ(b), with b ∈ U(q)?

2. (5 + 3) With notation as in Problem 6.5, but with m an odd prime number q, show that
a) every ρ ∈ R has ρ = ∑ 0 ≤ j ≤ q − 2 bjωj with unique bj ∈ Z. Hint: Problem 7.6 and
Proposition M.
b) every ρ ∈ R has ρ = ∑ 1 ≤ j ≤ q − 1 cjωj with unique cj ∈ Z.
3. (6) Evaluate the Legendre symbol ⎟⎠
⎞⎜⎝
⎛
43
11 without using quadratic reciprocity.
4. (2 + 4) Let Γ(s) be the gamm ...Read More

Solution Preview - h χ(b) ≠ 1 (which exists as χ ≠ χ0). Then (1 − χ(b)) ∑ a mod q χ(a) = ∑ a mod q χ(a) − χ(b)∑ a mod q χ(a) = ∑ a mod q χ(a) − ∑ a mod q χ(b)χ(a) = ∑ a mod q χ(a) − ∑ a mod q χ(ba) = 0, since aaba is a bijection U(q)U(q) (or, by Theorem 6.13). Now multiply both sides by the inverse of (1 − χ(b)) to ge

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