The
ring of
quaternions, discovered by
W.R. Hamilton provides the most often cited example of a
skew field or
division ring. It consists of all possible
sums of the form:
a1 +
bi +
cj +
dk
where
a,b,c and
d are
real numbers and
1,i,j and
k are members of the
quaternionic group. For concreteness we can take
1,i,j and
k to be 4 x 4 matrices:
1 = 1 0 0 0 i = 0 -1 0 0
j = 0 0 -1 0 k = 0
0 0 -1
0 1 0 0 1
0 0 0 0
0 0 1 0
0 -1 0
0 0 1 0 0
0 0 -1 1 0
0 0 0
1 0 0
0 0 0 1 0
0 1 0 0
-1 0 0 1
0 0 0
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combinatorics--