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Let p be a prime of the form 4k + 1

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Let p be a prime of the form 4k + 1. Wilson's theorem states that a 2 ≡ p − 1 (mod p), where a = (2k)!.

a) Show that p|(a + i)(a − i) in Z[i];

b) Show that p - a + i in Z[i] and p - a − i in Z[i];

c) Show that p is not irreducible in Z[i]

d) Let the factorization into irreducibles of p be z1z2 . . . zk. By considering the Euclidean function f(x + iy) = x 2 + y 2 = |x + iy| 2 , show that there must be exactly two irreducible factors of p.

e) Show that the factors must be of the form a + bi and a− bi for some a and b in Z (i.e. the two factors must be complex conjugates).

f) Deduce that p may be expressed as a sum of two square numbers.

g) Show that if q is a prime of the form 4k + 3, then it cannot be expressed as a sum of two squares. [Hint: what are the possible values of n 2 mod 4?]