Question

Irreducible elements in ring of gaussian integers

Answer 1

Answer:

Let π∈Z[i] irreducible. As you know N(π)≠1, so N(π) is a product of primes in Z>0.

If N(π) is prime, you are done.

Suppose N(π) is not prime.

If there is a prime p≡3(mod4) such that p∣N(π)=ππ¯ then p∣π, so π and p are associates in Z[i] and therefore N(π)=N(p)=p2. (Here I've used that any prime p≡3(mod4) is prime in Z[i].)

Otherwise, N(π) is a product of primes p≡1(mod4). Let p be such a prime. We also know that p=zz¯ where z∈Z[i] is a prime element. Then z∣ππ¯ and therefore π (or π¯) and z are associates, so N(π)=N(z)=p, a contradiction.

Step-by-step explanation:

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