I wonder an identity for the quaternions. Let $q_1$ and $q_2$ are two integral quaternions.

When is the term $\frac{q_1}{q_2}$ is integral quaternions.

Does it hold if and only if $N(q_1)\mid N(q_2)$?

Thank you.

  • $\begingroup$ "$q_1|q_2$" isn't a term, it's a relation. Do you mean "when is $\frac{q_1}{q_2}$ an integral quaternion, assuming $q_1$ and $q_2$ are?" $\endgroup$ – rschwieb May 3 '18 at 12:47
  • $\begingroup$ Yes. I mean when the term $\frac{q_1}{q_2}$ is a quaternion? Is there any property for this fraction? @rschwieb. $\endgroup$ – MATIRMAK May 3 '18 at 12:49
  • $\begingroup$ If that is the case, then I guess you want $N(q_2)|N(q_1)$ rather than the current order. $\endgroup$ – rschwieb May 3 '18 at 12:52
  • $\begingroup$ If $N(q_2)\mid N(q_1)$ holds, does it give that the term $\frac{q_1}{q_2}$ is integral quaternion? $\endgroup$ – MATIRMAK May 3 '18 at 12:55
  • $\begingroup$ Are you additionally assuming the quotient $N(q_1)/N(q_2)$ is an integer? Because as long as they are nonzero, they will always divide. $\endgroup$ – rschwieb May 3 '18 at 12:58

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