Results 1 to 2 of 2

Math Help - Relatively Prime Quadratic Integers

  1. #1
    Member
    Joined
    Jun 2010
    From
    United States
    Posts
    200

    Lightbulb Relatively Prime Quadratic Integers

    Hello Math Help Forum,

    I saw the following problem on another forum but the responses were all over the place and I was hoping if someone could give some clarity. The problem states:

    Assume 32 = \alpha\beta for \alpha,\beta relatively prime quadratic integers in Q[i]. It can be shown that \alpha = \epsilon \gamma^2 for some unit \epsilon and some quadratic integer \gamma in Q[i].

    Can someone explain how this is shown?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Well, 32 = 2^5 = (1+i)^5(1-i)^5. And 1+i,1-i are irreducible. But (1+i)=i(1-i). Hence 32 = i(1-i)^{10} is the unique expression of 32 in \mathbb{Z}[i] as a product of irreducibles. Can you finish from there?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Relatively Prime Quadratic Integers
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: September 17th 2010, 05:59 PM
  2. Relatively Prime Set of Integers
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: May 15th 2010, 01:32 AM
  3. Replies: 3
    Last Post: October 21st 2009, 08:58 PM
  4. Relatively prime quadratic integers
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: October 28th 2008, 12:11 AM
  5. relatively prime integers....
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: March 16th 2008, 02:00 PM

Search Tags


/mathhelpforum @mathhelpforum