Undergraduate seminar homework #3: Quaternions (ii)

To be handed in by Tuesday, April 7, 1998.

We define generalized quaternions having a basis 1, I, J, K over the reals, with addition and multiplication defined to be associative, distributive etc, and the multiplication satisfies The conjugate of a quaternion a = x0+ x1 I + x2 J + x3K is defined as a* = x0 - x1 I - x2 J - x3K.
The norm is N(a) = aa* = x02 + x12 + 3 (x22 + x32), and the trace is Tr(a) = a + a* = 2x0.

The integral quaternions are defined as A={ a0 +a1 I + a2 (I+J)/2 + a3 (1+K)/2 : all ai integers }.
  1. Show that A is closed under addition and multiplication and that for a in A, Tr(a) and N(a) are integers.

  2. Show that A has a Euclidean algorithm: Given a, b in A , b nonzero, there are q,r in A with
    a = qb + r and N(r) < N(b).
  3. We say that a divides c (on the right) if c = ab, a,b,c in A. Show that if m is a nonzero rational integer, dividing ab so that gcd(a,m)=1, then m divides b.
  4. i) Show that if p is a rational prime, then there are integers u, v so that u2 +v2 + 3 = 0 mod p.

    ii) Show that any rational prime p is a norm in A: p=N(a), for some a in A. Deduce that we can find integers x,y,z,w, with 4p = x2 +y2 +3(z2 + w2).

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