Show that Q_F is not a division ring.

[HaLAA]

Homework Statement

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.

Homework Equations

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+2jWith 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+2kp=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+kp=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.

 

[andrewkirk]

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.