# Number Theory Problems

## Question

1. For which primes p is - 2 a square mod p. Give the answer in terms of specific congruences p must satisfy.
2. For which primes p is -3 a square mod p. Give the answer in terms of specific congruences p must satisfy.
3. For which primes p is at least one of -1, 2 and -2 a square mod p. Give the answer in terms of specific congruences P must satisfy.
4. Let C be one of C₆, i√2, i, √2, √3 and ϕ. For each of the primes p below and each of the values of C listed, say whether ±p is in the image of the norm map N: Z[C] -> 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.

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