Results 1 to 2 of 2

Math Help - Find units in ring of integers

  1. #1
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2

    Find units in ring of integers

    Let R be the ring of integers in  \mathbb {Q} [ \sqrt {-3} ] . Determine the units in R.

    Let  x = a + b \sqrt {-3}, \ \ \ a,b \in \mathbb {Z} be a unit of R, then the norm of x, that is,  N_{R} (x) = 1,-1

     N_{R} (x) = N_{R} (a+b \sqrt {-3} ) = a^2 + b^2 (-3) = =1, -1

    I can only think of +1 and -1 being units in R, anything more?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member JaneBennet's Avatar
    Joined
    Dec 2007
    Posts
    293
    A unit in a ring with multipicative identity 1 is a divisor of 1; if the ring is also a normed vector space, then the norm of a unit must be 1.

    So a+b\sqrt{-3} is a unit in \mathbb{Q}\left[\sqrt{-3}\right]\ \Rightarrow\ a^2+3b^2=1..

    Conversely, any element of the form a+b\sqrt{-3} where a^2+3b^2=1 is a unit in \mathbb{Q}\left[\sqrt{-3}\right], its mutliplicative inverse being a-b\sqrt{-3}:


    \left(a+b\sqrt{-3}\right)\left(a-b\sqrt{-3}\right)\ =\ 1


    Hence the units of \mathbb{Q}\left[\sqrt{-3}\right] are \left\{a+b\sqrt{-3}:a,b\in\mathbb{Q},a^2+3b^2=1\right\}.

    By the way, there are units other than 1: for example, \left(\frac{1+\sqrt{-3}}{2}\right)\left(\frac{1-\sqrt{-3}}{2}\right)=1.
    Last edited by JaneBennet; February 26th 2008 at 02:38 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Units in a polynomial ring
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: September 2nd 2010, 05:52 AM
  2. Units in a ring?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 29th 2010, 02:05 AM
  3. Units of a ring
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 2nd 2010, 07:01 PM
  4. Units in Gauss integers ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 1st 2008, 03:33 PM
  5. Units of Ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 11th 2006, 01:59 PM

Search Tags


/mathhelpforum @mathhelpforum