Question 1

Factor the number 7! as a product of Gaussian primes:

Question 2

How many Gaussian primes are there of norm smaller than 11?

How many Gaussian integers are there of norm 120?

Question 3

a) Suppose that N(z)=p^2 for a Gaussian integer z and a prime number p. Show that if p=4k+3, then z is a Gaussian prime. Show that if p=4k+1, then z is not a Gaussian prime.

b) Prove the following criterion: a Gaussian integer is a Gaussian prime if and only if N(z) is a prime or a square of a prime of the form p=4k+3.