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
- I2 = -1, J2 = -3, and so
K2 = -3.
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 }.
- 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 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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