Undergraduate seminar homework #2: Quaternions
To be handed in by Tuesday, March 31, 1998.
 Show that conjugation of quaternions satisfies
(ab)^{*} = b^{*} a^{*}

Denote by H the "maximal order" of integer quaternions
a_{0}p +a_{1}i + a_{2} j+
a_{3} k, p = (1+i+j+k)/2,
a_{i} integers.
Show that if a quaternion x is in H then both tr(x) = a + a^{*}
and N(x) = a a^{*} are integers.

For the following integer quaternions a, b in H, find q,r in H so that
a=qb+r an N(r) < N(b):
i) a = 7p+j+3k, b= 3; (ii) a=2+i, b=1+k.

Let Q_{0} be the quaternions with integral coefficients,
a_{0} +a_{1}i + a_{2} j+ a_{3} k, all
a_{i} integers. Find an example of a, b in Q_{0} for
which there are no q, r in Q_{0} with a=qb+r and N(r) < N(b).
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