Results 1 to 2 of 2

Thread: Ideals of the Gaussian Integers

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    128

    Ideals of the Gaussian Integers

    Hello!

    Given the Gaussian Integers $\displaystyle Z[i] = \{ a + bi | a,b \in Z\}$

    Then I am trying to show that

    1) $\displaystyle I = \{ a + bi\ |\ a,b \in Z, 5|a, 5|b\}$ is an ideal in $\displaystyle Z[i]$ but $\displaystyle I$ is not maximal. To be clear, I mean 5 divides a, and 5 divides b.

    To show it's an ideal, I have to show that it "absorbs" any element from Z[i] "into"
    it... seems like something that should follow from number theory. Also, I am not sure how it is not maximal! I need to find some other ideal that I is properly contained in? I can't think of any!

    2) $\displaystyle J = <2+ i>$ is NOT a prime ideal in $\displaystyle Z[i]$ (<> refers to the principal ideal)...

    and then I am trying to figure out how many elements there are in the factor ring $\displaystyle \frac{Z[i]}{J}$

    Any guidance or assistance appreciated! Thanks!!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    For part one, you have to show that for any element $\displaystyle z=x+yi\in \mathbb{Z}[i]$, $\displaystyle zI\subseteq I$. Did you try multiplying $\displaystyle (x+yi)(a+bi)$, and checking the condition?

    To finish that problem, try showing that $\displaystyle I\subset \langle 1+i\rangle$.

    For part two, if you can decide what $\displaystyle \mathbb{Z}[i]/J$ is, then you can resolve your question about the primality of $\displaystyle J$. (Recall that an ideal $\displaystyle I$ is a prime ideal if and only if $\displaystyle R/I$ is an integral domain.)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Gaussian integers.
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Apr 6th 2011, 02:17 AM
  2. Replies: 7
    Last Post: Aug 3rd 2010, 01:31 PM
  3. Gaussian Integers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: May 23rd 2009, 06:31 PM
  4. Gaussian integers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Jun 5th 2008, 08:44 AM
  5. Gcd in Gaussian integers
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: Jan 2nd 2007, 10:30 AM

Search Tags


/mathhelpforum @mathhelpforum