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Math Help - Irreducible in Z[i]

  1. #1
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    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?
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    Quote Originally Posted by tttcomrader View Post
    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.
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  3. #3
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    Quote Originally Posted by tttcomrader View Post
    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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