Buy math 324 assignment 8 solution alberta university
Question - Math 324 Assignment 8 (due November 12)
1. (6) If Ï‡ is a non-principal Dirichlet character mod q, show that âˆ‘ a mod q Ï‡(a) = 0.
Hint: What happens to âˆ‘ a mod q Ï‡(a) if it is multiplied by Ï‡(b), with b âˆˆ U(q)?
2. (5 + 3) With notation as in Problem 6.5, but with m an odd prime number q, show that
a) every Ï âˆˆ R has Ï = âˆ‘ 0 â‰¤ j â‰¤ q âˆ’ 2 bjÏ‰j with unique bj âˆˆ Z. Hint: Problem 7.6 and
b) every Ï âˆˆ R has Ï = âˆ‘ 1 â‰¤ j â‰¤ q âˆ’ 1 cjÏ‰j with unique cj âˆˆ Z.
3. (6) Evaluate the Legendre symbol âŽŸâŽ
11 without using quadratic reciprocity.
4. (2 + 4) Let Î“(s) be the gamm
a function of D.3. Show that a) Î“(1) = 1, and
b) Î“(s +1) = sÎ“(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 Ï‡(b) â‰ 1 (which exists as Ï‡ â‰ Ï‡0). Then (1 âˆ’ Ï‡(b)) âˆ‘ a mod q Ï‡(a) =
âˆ‘ a mod q Ï‡(a) âˆ’ Ï‡(b)âˆ‘ a mod q Ï‡(a) = âˆ‘ a mod q Ï‡(a) âˆ’ âˆ‘ a mod q Ï‡(b)Ï‡(a) = âˆ‘ a mod q Ï‡(a) âˆ’
âˆ‘ a mod q Ï‡(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 âˆ’ Ï‡(b)) to ge
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