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Proving an element of Z[i] is irreducible?

cliveb2016

As above (I didn't want to type it out again so hopefully this is clear from the picture.)

Any help with 2?

Reply 1

Gregorius

Original post by cliveb2016

Any help with 2?

If x is irreducible in the Gaussian integers, then N(x) is prime in the ordinary integers. Can you take it from there, using the first part?

Reply 2

cliveb2016

Original post by Gregorius

If x is irreducible in the Gaussian integers, then N(x) is prime in the ordinary integers. Can you take it from there, using the first part?

Suppose a is irreducible in Z, then N(a) is prime in Z but N(a)=pq which is not prime in Z (contradiction) so a is not irreducible in Z and so a cannot be written as a product of 1 irreducible so it must be written as a product of more than one but strictly less than 2 irreducibles hence it must be the product of exactly 2 irreducibles.

Is that right?

I got stuck since I didn't know that
If x is irreducible in the Gaussian integers, then N(x) is prime in the ordinary integers.

I thought that implication is only true the reverse way i.e

N(x) is prime in the ordinary integers then x is irreducible in the Gaussian integers.

Could you explain why your statement holds. (I tried to prove with no luck)

Reply 3

Gregorius

Original post by cliveb2016

Could you explain why your statement holds. (I tried to prove with no luck)

It doesn't - I got it wrong!

So, let's try again: if x = a+ ib is a Gaussian integer, with a and b non-zero, then x irreducible implies N(x) prime. So you can apply the reasoning you have already worked out. Then you have to deal with the either a or b equals to zero cases.

Sorry about that.

Reply 4

cliveb2016

Original post by Gregorius

No worries thanks for the help.

I think I can finish the proof with the claim "if x = a+ ib is a Gaussian integer, with a and b non-zero, then x irreducible implies N(x) prime."

Would you be able to elaborate why this is true?

thanks

Reply 5

Gregorius

Original post by cliveb2016

Would you be able to elaborate why this is true?

Getting into the realms of bookwork - page 69 of Baker's "Concise Introduction", for example. Did this question arise from an algebraic number theory course or has it been sprung on you in a lower level algebra course?