Thread: Irreducible in Z[i]

Show that is an irreducible in .

Proof. Suppose that 1 - i = ab, for a and b in Z[i], we need to show that either a or b is a unit.

I'm having trouble trying to do this one, is there a certain factor that 1 - i has that can allow me to conclude the other has to be a unit?

Originally Posted by tttcomrader

Show that is an irreducible in .

Proof. Suppose that 1 - i = ab, for a and b in Z[i], we need to show that either a or b is a unit.

I'm having trouble trying to do this one, is there a certain factor that 1 - i has that can allow me to conclude the other has to be a unit?

Hint: where . Now equate.

Originally Posted by tttcomrader

Show that is an irreducible in .

Proof. Suppose that 1 - i = ab, for a and b in Z[i], we need to show that either a or b is a unit.

I'm having trouble trying to do this one, is there a certain factor that 1 - i has that can allow me to conclude the other has to be a unit?

Have you learned about the norm in Z[i]? If so, you should use the fact that an element with prime norm is irreducible.