1. ## Irreducible in Z[i]

Show that $1 - i$ is an irreducible in $Z[i]$.

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?

Show that $1 - i$ is an irreducible in $Z[i]$.

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: $1+i = (a+ib)(x+iy)$ where $a,b,x,y\in \mathbb{Z}$. Now equate.

Show that $1 - i$ is an irreducible in $Z[i]$.

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.

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# show that 1 i is an irreducible element of z(i)

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