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

IJ=  JI =K, JK = KJ = 3I, KI = IK = J
 I^{2} = 1, J^{2} = 3, and so
K^{2} = 3.
The conjugate of a quaternion
a = x_{0}+ x_{1} I + x_{2} J +
x_{3}K is defined as
a^{*} = x_{0}  x_{1} I  x_{2} J 
x_{3}K.
The norm is
N(a) = aa^{*} =
x_{0}^{2} + x_{1}^{2} + 3
(x_{2}^{2} + x_{3}^{2}),
and the trace is
Tr(a) = a + a^{*} =
2x_{0}.
The integral quaternions are defined as
A={ a_{0} +a_{1} I + a_{2} (I+J)/2 +
a_{3} (1+K)/2 : all a_{i} integers }.
 Show that A is closed under addition and multiplication and that
for a in A, Tr(a) and N(a) are
integers.

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).

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.

i) Show that if p is a rational prime, then there are integers u, v so
that u^{2} +v^{2} + 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 = x^{2} +y^{2} +3(z^{2} + w^{2}).
Back to
Undergraduate Seminar Homepage.