Math 425-525 Exam 2 Spring 2015 Please submit your solution in word processed form. Do not use any calculations other than powers, mods, and explicit multiplications of algebraic integers in a computer algebra package. 1. For which primes p is 2 a square mod p. Give the answer in terms of specic congruences p must satsify. 2. For which primes p is 3 a square mod p. Give the answer in terms of specic congruences p must satsify. 3. For which primes p is at least one of 1, 2 and 2 a square mod p. Give the answer in terms of specic congruences p must satsify. 4. Let be one of 6 , i 2, i, 2, 3 and . For each of the primes p below and each of the values of listed, say whether p is in the image of the norm map N : Z[] Z. (a) p = 2 (b) p = 3 (c) p = 2659 (d) p = 4079 5. Let p = 2659. Explicitly write p as a norm for two of the values of uncovered in problem 4