Results 1 to 2 of 2

Thread: proving an isomorphism

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    38

    proving an isomorphism

    Hi, could you please help me with the following question.


    How do i prove that the set of all complex numbers of the form a + bi with a, b in Q (rationals) is isomorphic to the field of fractions of the ring Z[i] of Gaussian integers?

    Thanks for your help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Louise View Post
    Hi, could you please help me with the following question.


    How do i prove that the set of all complex numbers of the form a + bi with a, b in Q (rationals) is isomorphic to the field of fractions of the ring Z[i] of Gaussian integers?

    Thanks for your help.
    looking at them as subsets of $\displaystyle \mathbb{C},$ they're more than just isomorphic. they are basically equal. because an element of the field of fractions of $\displaystyle \mathbb{Z}[i]$ is in the form $\displaystyle x=\frac{a+bi}{c+di},$ where $\displaystyle a,b,c,d \in \mathbb{Z}$

    and $\displaystyle cd \neq 0.$ but we have $\displaystyle x=\frac{(a+bi)(c-di)}{c^2+d^2}=\frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i \in \mathbb{Q}(i).$ on the other hand if $\displaystyle y \in \mathbb{Q}(i),$ then $\displaystyle y=p+qi,$ for some $\displaystyle p=\frac{a}{b}, \ q=\frac{c}{d},$ with $\displaystyle a,b,c,d \in \mathbb{Z},$ and $\displaystyle bd \neq 0.$ then

    $\displaystyle y=\frac{ad + bc i}{bd}.$ clearly both $\displaystyle ad + bc i$ and $\displaystyle bd$ are in $\displaystyle \mathbb{Z}[i]$ and thus $\displaystyle y$ is in the field of fractions of $\displaystyle \mathbb{Z}[i]$ because $\displaystyle bd \neq 0.$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving isomorphism to Zn
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Nov 20th 2011, 08:04 PM
  2. [SOLVED] What isomorphism do I need?
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: Jun 2nd 2011, 04:00 AM
  3. Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: Oct 27th 2010, 12:08 AM
  4. Replies: 4
    Last Post: Feb 14th 2010, 03:05 AM
  5. Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Mar 8th 2009, 03:01 PM

Search Tags


/mathhelpforum @mathhelpforum