Undergraduate seminar homework #2: Quaternions

To be handed in by Tuesday, March 31, 1998.

  1. Show that conjugation of quaternions satisfies (ab)* = b* a*
  2. Denote by H the "maximal order" of integer quaternions a0p +a1i + a2 j+ a3 k, p = (1+i+j+k)/2, ai integers. Show that if a quaternion x is in H then both tr(x) = a + a* and N(x) = a a* are integers.
  3. 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.
  4. 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).

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