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Show that Q_F is not a division ring.

  1. Jun 4, 2015 #1
    1. The problem statement, all variables and given/known data
    Let F be a finite field of characteristic p∈{2,3,5}. Consider the quaternionic ring, Q_F={a_1+a_i i+a_j j+a_k k|a_1,a_i,a_j,a_k ∈ F}. Prove that Q_F is not a division ring.

    2. Relevant equations


    3. The attempt at a solution
    Let α=1+i,β=1+i+j∈QF. Then

    αβ=(1+i)(1+i+j)=(1−1)+(1+1)i+(1+1)j+(1−1)k=2i+2j


    With characteristic p=2, αβ=0.

    With characteristic p=3, αβ=2(i+j).

    With characteristic p=5, αβ=3(i+j).



    As I keep working with the method I have the following:


    αγ=(1+i)(i+2j)=(−1)+(1)i+(2)j+(2)k=−1+i+2j+2k


    p=2, αγ=1+i. p=3, αγ=2+i+2j+2k. p=5, αγ=4+i+2j+2k.


    βγ=(1+i+j)(i+2j)=(−1−2)+(1)i+(2)j+(2−1)k=−3+i+2j+k


    p=2, βγ=1+i+k. p=3, βγ=i+2j+k. p=5, βγ=2+i+2j+k.

    I don't get any zero divisors, I may make some error somewhere because I should get zero divisors when p=3,5 also.
     
  2. jcsd
  3. Jun 6, 2015 #2

    andrewkirk

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    A division ring is a ring in which every non-zero element has a multiplicative inverse. So to prove that a ring is not a division ring, you need to find an element that has no multiplicative inverse.

    The manipulations you have done seem to be related to a different question, which is whether the ring has any zero divisors. Do you have a theorem that says the two questions are related?

    You are likely to receive more help if you use LaTeX to present the problem clearly. The first line that defines Q_F is too hard to decipher in the format you have used.
     
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