1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Finding the units of the ring Z[sqrt(-3)]

  1. Jun 11, 2009 #1
    1. The problem statement, all variables and given/known data

    In the ring [tex]Z[\sqrt{-3}][/tex] find all units and prove that 2 is irreducible but 7 is not.

    2. Relevant equations

    3. The attempt at a solution

    Well a unit is a non-zero element of the ring that when multiplied by some other non-zero element of the ring gives the unity of the ring.

    ie; [tex] ab = 1\;a,b \in Z[\sqrt{-3}] [/tex]

    in otherwords b is the multiplicative inverse of a.

    a is of the form [tex] a = x + y\sqrt{3} i [/tex] which just has the inverse of [tex]\frac{x - y\sqrt{3}i}{x^{2} + 3y^{2}}[/tex] (ie; b has this form)

    So everything with the form of b with integer coefficients is a unit and 0 is not a unit.

    I really don't understand what this question is asking, because I know what I've written down is completely stupid.

    How would one go about finding the units of this ring?
  2. jcsd
  3. Jun 11, 2009 #2


    User Avatar
    Science Advisor

    Which means that [itex]x^2+ 3y^2[/itex] is a factor of both x and y. For what x and y is that true?

    Another way to do this is to look at [itex](x+ y\sqrt{3})(a+ b\sqrt{3})[/itex][itex]= (ax+ 3by)+ (ay+bx)\sqrt{3}= 1[/itex]. What must x, y, a, and b be?

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook