Results 1 to 2 of 2

Math Help - Gaussian Integers

  1. #1
    Super Member Deadstar's Avatar
    Joined
    Oct 2007
    Posts
    722

    Gaussian Integers

    In this part, you may assume any facts about the factorisation theory of Z[i], the ring of Gaussian Integers, and of Z provided that you state clearly the properties that you are using.
    Let p be an integer prime for which there is an element a in Z with a^2 + 1 = p. Write down a factorisation of p in Z[i] and show that it is a proper factorisation (that is, neither factor is a unit of Z[i]). Deduce that p is not a prime in the Gaussian integers. Show that the factors of p that you obtained are primes in Z[i].

    Thoughts...

    p = (a + i)(a - i)
    I'm not certain about this but i think that a prime cant be a unit? So in Z that would be \pm 1?
    If so then a \neq 0 hence since the only units in Z[i] are 1,-1,i ,-i. Neither factor is not a unit, i.e. proper.

    p is not prime since p doesn't divide either factor. This is because a \pm i is imaginary and p is real.

    Showing a \pm i are primes in Z[i]...
    a + bi is prime if a^2 + b^2 is an integer prime of the form 4k+1. So how do i show a^2 + 1 is of the form 4k+1?

    Actually a thought has just occurred... Since a^2 + 1 is prime, a^2 has to be even, i.e. a is even. So let a be of the form 2n. Then we have (2n)^2 + 1 = 4k + 1 so 4n^2 = 4k, hence k = n^2? So... ?

    One final note... a^2 + 1 could be even, but only if a=1, since 2 is the only even prime. But 1+i and 1-i are primes in Z[i] so that takes care of the a=odd case.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Deadstar View Post
    In this part, you may assume any facts about the factorisation theory of Z[i], the ring of Gaussian Integers, and of Z provided that you state clearly the properties that you are using.
    Let p be an integer prime for which there is an element a in Z with a^2 + 1 = p. Write down a factorisation of p in Z[i] and show that it is a proper factorisation (that is, neither factor is a unit of Z[i]). Deduce that p is not a prime in the Gaussian integers. Show that the factors of p that you obtained are primes in Z[i].

    Thoughts...

    p = (a + i)(a - i)
    I'm not certain about this but i think that a prime cant be a unit? So in Z that would be \pm 1?
    If so then a \neq 0 hence since the only units in Z[i] are 1,-1,i ,-i. Neither factor is not a unit, i.e. proper.

    p is not prime since p doesn't divide either factor. This is because a \pm i is imaginary and p is real.

    Showing a \pm i are primes in Z[i]...
    a + bi is prime if a^2 + b^2 is an integer prime of the form 4k+1. So how do i show a^2 + 1 is of the form 4k+1?

    Actually a thought has just occurred... Since a^2 + 1 is prime, a^2 has to be even, i.e. a is even. So let a be of the form 2n. Then we have (2n)^2 + 1 = 4k + 1 so 4n^2 = 4k, hence k = n^2? So... ?

    One final note... a^2 + 1 could be even, but only if a=1, since 2 is the only even prime. But 1+i and 1-i are primes in Z[i] so that takes care of the a=odd case.
    Define N: \mathbb{Z}[i]\to \mathbb{Z}[i] by N(\alpha) = \alpha \bar \alpha , so if \alpha=a+bi then N(\alpha) = a^2+b^2.
    It is easy to show that \alpha is a unit if and only if N(\alpha) = 1.
    To show that a\pm 1 is not a unit just notice that N(a\pm 1) = (a\pm 1)^2 > 1 (if |a|> 1).

    If N(\alpha) = p for a prime then \alpha must be prime. Because otherwise \alpha = \beta \gamma for non-units and so p=N(\alpha) = N(\beta)N(\gamma). Which is a problem because p is prime.
    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: April 6th 2011, 03:17 AM
  2. Fields and gaussian integers
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 9th 2010, 07:44 PM
  3. Replies: 7
    Last Post: August 3rd 2010, 02:31 PM
  4. Gaussian integers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: June 5th 2008, 09:44 AM
  5. Gcd in Gaussian integers
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: January 2nd 2007, 11:30 AM

Search Tags


/mathhelpforum @mathhelpforum