Chapter 5

Question 1

Lemma V.2.1 irreducibles in the gaussian integers

Answer 1

Suppose that p is an integer, p>0 is irreducible.

(i) if p=2 then p is an associate to (1+i)² and (1+i) is irreducible in the gaussian integers
(ii) if p≡3mod4 then p is irreducible in the gaussian integers
(iii) if p≡1mod4 then p is not irreducible in the gaussian integers and p = aa* for some irreducible a in the gaussian integers. Moreover, N(a)=p.

Question 2

The list of irreducibles in the gaussian integers (L)

Answer 2

• The associates of 1+i
• The associates of p where p is is a positive irreducible in the integers and p≡3mod4
• The irreducible factors of p where p is a positive irreducible in the integers with p≡1mod4

Question 3

Theorem V.2.2 L is complete

Answer 3

L is a complete list of the irreducibles in the gaussian integers

Question 4

Theorem V.3.1 (fermats theorem)

Answer 4

n is a sum of two squares iff each βᵢ is even

Question 5

We can write n as

Answer 5

n = 2^α₀p₁^α₁ …pᵣ^αᵣq₁^β₁…qₜ^βₜ

where the ps are distinct primes congruent to 1 mod4 and the qs are distinct primes congruent to 3mod4