MAT-115A PROBLEM SET 9

MAT-115A PROBLEM SET 9

(Due July 30, Wednesday)

1. Evaluate the following values of Legendre symbols: (1) (3/53); (2) (7/79); (3) (29/31); (4) (31/641).

2. Evaluate the Legendre symbols (503/773) and (501/773).

3. Find a criterion for the primes p such that (5/p) = 1.

4. Let p > 3 be a prime number. Show that x2 ≡ −3(mod p) is solvable i� p ≡ 1(mod 6).

5. Prove the following: if p = 4k +1 and d∣k, then (d/p) = 1.

6. Let p be a prime number. We call a unit a in Z/pZ a primitive root, if ordp(a) = p − 1, i.e. any unit in Z/pZ can be written as some power of a. If p is of the form 2n +1, prove that the primitive roots in Z/pZ are precisely the quadratic non-residues modulo p. If n > 1, prove 3 is always a primitive root.

7. Let p be an odd prime, and (a,p) = 1. Show that if x2 = a(mod p) has solutions, then x2 = a(mod pN) always has solutions, for any N > 1.

8. Does x2 +x+1 ≡ 0(mod 997) have solutions? Why or why not?

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