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Factoring in the Gaussian Integers

  1. Mar 3, 2012 #1
    I need to factorise 70 into primes, how do I go about this?

    So far I have 2,5,7 as primes in Z.

    So I suppose I need to factorise these in Z?

    2 = (1+i)(1-i)

    How do I go around doing the other two, is it possible that they're primes in Z?

    Edit:

    I have a corollary where if p is a prime in Z, then p is a prime in Z if p = 3 mod 4

    So 7 stays prime in Z.

    Also I have that if the norms of elements in Z are congruent to 1 mod 4 and prime in Z, then the elements in Z are prime, so

    5 = (2+1)(2-i) = 1 mod 4

    so 70 = 7*(2+1)(2-i)*(1+i)(1-i)

    Correct?

    Thanks
     
    Last edited: Mar 3, 2012
  2. jcsd
  3. Mar 8, 2012 #2

    morphism

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    Yes, that's correct.
     
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