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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
Proposition M.
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
âŽžâŽœâŽ
âŽ›
43
11 without using quadratic reciprocity.
4. (2 + 4) Let Î“(s) be the gamm ...Read More

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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