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
a0p +a1i + a2 j+
a3 k, p = (1+i+j+k)/2,
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 Q0 be the quaternions with integral coefficients,
a0 +a1i + a2 j+ a3 k, all
ai integers. Find an example of a, b in Q0 for
which there are no q, r in Q0 with a=qb+r and N(r) < N(b).
Undergraduate Seminar Homepage.