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Math Help - Unit

  1. #1
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    Unit

    Hello Forum! I have the following problem, which is related to the ring of quaternions, I tried but I had no luck in making it, I hope I can help:


    Let I be the ring of integers Hamilton quaternions and define:



    N : I\rightarrow{}Z\ with\ N(a+bi+cj+dk)=a^2+b^2+c^2+d^2


    (The N is called norm)


    Prove that an element of R is a unit if and only if it has norm +1. Addition show that I^x is isomorphic to the quaternion group of order 8.


    Thanks
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  2. #2
    Member ModusPonens's Avatar
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    Re: Unit

    I don't understand the question. the quaternions form a division ring, which means every element except zero has inverse, and thus is a unit.
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  3. #3
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    Re: Unit

    ModusPonens, I think it's because I is the quaternions with integer coordinates. It is similar to the Gaussian integers.

    Suppose that x is a unit of I. That is, there exists a y\in I such that xy = yx = 1_I, where 1_I = 1 is the identity of I. Next, N is a map from I into the non-negative integers. You should prove that N(ab) = N(a)N(b) where the multiplication on the LHS is in I and the multiplication on the RHS is in \mathbb{Z}. Then you have

    N(xy) = N(x)N(y) = N(1) = 1

    Can you take it from here?
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  4. #4
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    Re: Unit

    he means, presumably, the ring of quaternions with integer coefficients.

    what you need to do is show the norm N is multiplicative. that is, for integral quaternions q,q', N(qq') = N(q)N(q').

    then if q is a unit in R, we have qp = 1, for some integral quaternion p. so N(q)N(p) = N(qp) = N(1) = 1.

    now N(q) and N(p) are integers, so they are units of Z, so N(q) = 1 or -1, since those are the only units of Z.

    but N(q) ≥ 0, so N(q) = 1.

    this means that precisely one of a,b,c, or d is 1, that is the units of R are:

    {1,-1.i,-i,j,-j,k,-k}.
    Last edited by Deveno; August 23rd 2012 at 11:50 PM.
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