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:
(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 is isomorphic to the quaternion group of order 8.
August 23rd 2012, 08:08 PM
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.
August 23rd 2012, 11:44 PM
ModusPonens, I think it's because is the quaternions with integer coordinates. It is similar to the Gaussian integers.
Suppose that is a unit of . That is, there exists a such that , where is the identity of . Next, is a map from into the non-negative integers. You should prove that where the multiplication on the LHS is in and the multiplication on the RHS is in . Then you have
Can you take it from here?
August 23rd 2012, 11:47 PM
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: