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# Alberta university math 324 homework solution

Question - Math 324 Assignment 9 (due November 19)

1. (6) Evaluate the Legendre symbol âŽŸâŽ
âŽžâŽœâŽ
âŽ›
991
667 by the method of Section 11.2

2. (6) If k â‰¥ 3, show that there are exactly 4 solutions mod 2k of the congruence
x2 â‰¡ 1 mod 2k. Hint: Problem 7.5.

3. (6) Find a congruence describing all odd primes for which 13 is a quadratic residue.

4. (4 + 4) Determine the number of solutions mod m of the congruences
a) x2 âˆ’ 3x + 1 â‰¡ 0 mod 1073, and
b) x2 + x + 2 â‰¡ 0 mod 1219.

5. (3 + 5) Show that, for all odd primes p,
a) S:= {a + bi: a, b in Z} is a subring of C with Z âˆ© pS âŠ† pZ, and
b) prove Theorem 11.6 by the method of Section G. Hint: Consider the element 1 + i âˆ ...Read More

Solution Preview - ) is prime since this is a Legendre symbol. From 667 = 23â‹…29, with 23 (â‰¡ âˆ’ 1 mod 4) and 29 (â‰¡ 1 mod 4) both prime, we have âŽŸâŽ  âŽžâŽœâŽ âŽ› 991 667 = âŽŸâŽ  âŽžâŽœâŽ âŽ› 991 23âŽŸâŽ  âŽžâŽœâŽ âŽ› 991 29 = âˆ’ âŽŸâŽ  âŽžâŽœâŽ âŽ› 23 991âŽŸâŽ  âŽžâŽœâŽ âŽ› 29 991 = âˆ’ âŽŸâŽ  âŽžâŽœâŽ âŽ› 23 2âŽŸâŽ  âŽžâŽœâŽ âŽ› 29 5 = (âˆ’ 1)âŽŸ

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