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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 a function of D.3. Show that a) Γ(1) = 1, and
b) Γ(s +1) = sΓ(s) for all s > 0. Hint: Integration by parts.

5. (6)(11.1 #44) Let p, q be odd primes with q = 4p + 1. Show that 2 is a primitive root
mod q. Hint: Theorem 11.3 and 11.6.

6. (8)(4.1, #46) Five men and a monkey are shipwrecked on an island. The men have
collected a pile of coconuts that they plan to divide equally among themselves the next
morning. Not trusting the other men, one of the group wakes up during the night and
divides the pile into five equal parts with one left over, which he gives to the monkey. He
then hides his portion of the pile. During the night, each of the other four men does
exactly the same thing by dividing the pile that he finds into five equal parts, leaving one
coconut for the monkey, and hiding his portion. In the morning, the men gather and split
the remaining pile into five equal parts and one is again left over for the monkey. What is
the minimum number of coconuts the men could have collected for their original pile? ...Read Less

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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