Alberta university math 324 homework solution

Question - Math 324 Assignment 9 (due November 19)

1. (6) Evaluate the Legendre symbol ⎟⎠
⎞⎜⎝
⎛
991
667 by the method of Section 11.2

2. (6) If k ≥ 3, show that there are exactly 4 solutions mod 2k of the congruence
x2 ≡ 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 − 3x + 1 ≡ 0 mod 1073, and
b) x2 + x + 2 ≡ 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 ∩ pS ⊆ pZ, and
b) prove Theorem 11.6 by the method of Section G. Hint: Consider the element 1 + i ∠...Read More ˆ S
as the analogue ‘G(2)’ of G(q), and compute (1 + i)p mod pS in the analogous two ways.

6. (6) Find all integer solutions x of 3x ≡ x mod 7. Hint: Since x is a solution ⇔ x + 42 is
a solution, it makes sense to determine all of the solutions mod 42; also observe that there
are no solutions x ≡ 0 mod 7. ...Read Less

Solution Preview - ) is prime since this is a Legendre symbol. From 667 = 23⋅29, with 23 (≡ − 1 mod 4) and 29 (≡ 1 mod 4) both prime, we have ⎟⎠ ⎞⎜⎝ ⎛ 991 667 = ⎟⎠ ⎞⎜⎝ ⎛ 991 23⎟⎠ ⎞⎜⎝ ⎛ 991 29 = − ⎟⎠ ⎞⎜⎝ ⎛ 23 991⎟⎠ ⎞⎜⎝ ⎛ 29 991 = − ⎟⎠ ⎞⎜⎝ ⎛ 23 2⎟⎠ ⎞⎜⎝ ⎛ 29 5 = (− 1)⎟

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