Results 1 to 5 of 5

Math Help - gaussian prime factorisation

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    32

    gaussian prime factorisation

    if you look at the wolframalpha calculation page:
    -13-26i - Wolfram|Alpha

    it factorises -13-26i as

    i(1+2 i)(2+3 i)(3+2 i)

    my question is why is the i a necessary factor to take out? are the rules regarding this?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2

    Re: gaussian prime factorisation

    Hint: (1-i) is NOT a Gaussian Prime. It can be factored: i(1+i)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485

    Re: gaussian prime factorisation

    Quote Originally Posted by TKHunny View Post
    Hint: (1-i) is NOT a Gaussian Prime. It can be factored: i(1+i)
    I don't think that this is correct. There is a characterisation of Gaussian primes that says that a Gaussian integer is prime if and only if one of the conditions is true:

    • Its norm is prime (as an integer)
    • It is a prime (as an integer) of the form 4k+3


    In this case, the norm of 1-i is 1^2+(-1)^2=2, which is prime. So 1-i is prime.

    The issue that should be brought up is that unique factorisation only holds up to a unit (i.e. invertible number). For example, normal integer factorisation follows this idea: 25=5\times 5 or 25=(-5)\times (-5). In the second example, we have multiplied the factors by -1 (which is invertible), but we do not consider it to be inherently different from the first. So in this case, 1-i only differs from 1+i by multiplying by the unit i, which is no difference at all, as far as unique factorisation goes.


    So to address original question, throwing in i is analogous to repeatedly multiplying by -1 with the usual integer factorisation. If it bothers you, you may let one of the terms absorb it.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2

    Re: gaussian prime factorisation

    I'm making stuff up again?! Only explanation I could think of for the question presented. It's -i, anyway. Wow. Taking a break...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485

    Re: gaussian prime factorisation

    No worries! Mistakes happen.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Need help understanding prime factorisation
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: February 20th 2011, 12:38 PM
  2. Worst case for prime factorisation
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: December 16th 2009, 01:04 AM
  3. prime factorisation
    Posted in the Algebra Forum
    Replies: 3
    Last Post: September 6th 2009, 06:06 AM
  4. Unique Prime factorisation
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: October 12th 2008, 03:08 PM
  5. Replies: 2
    Last Post: March 25th 2008, 07:38 AM

Search Tags


/mathhelpforum @mathhelpforum