Polynomials and Quaternions
Serge Ballif, January 24, 2008
The main reference for the talk is
A First Course in Noncommutative Rings
by T. Y. Lam.
1
Prerequisites and Definitions
1.1
Rings
Rings
Definition 1.
A
ring
R
is a set together with two binary operations “+” and “
×
” such that the following
three properties hold.
1. (
R,
+) form an additive group.
2. Multiplication is associative (i.e. (
ab
)
c
=
a
(
bc
) for all
a, b, c
∈
R
).
3. The left and right distributive laws hold:
a
(
b
+
c
) =
ab
+
ac
and
(
a
+
b
)
c
=
ac
+
bc.
Convention
For the purposes of this talk, we will assume that all rings have an identity element 1 such that 1
r
=
r
for
all
r
∈
R
.
Examples of Rings
Example
2
.
The following rings are familiar to all mathematicians.
•
— the real numbers
•
— the complex numbers
•
— the integers
•
Mat
n
(
) — the ring of
n
×
n
realvalued matrices
1.2
Division Rings
Division Rings
Definition 3.
A ring
R
is a
division ring
iff every nonzero element of
R
has a twosided inverse element iff
(
R
 {
0
}
,
×
) is a group.
Remark
A field is a commutative division ring.
Example
4
.
•
and
are division rings.
•
and Mat
n
(
) are not division rings.
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1.3
Quaternions
The Real Quaternions
Definition 5.
The ring of real quaternions
is a 4dimensional real vector space with basis
{
1
, i, j, k
}
.
is a division ring with multiplication structure determined by the relations
i
2
=
j
2
=
k
2
=

1 =
ijk.
An element
q
of
is of the form
q
=
a
+
bi
+
cj
+
dk,
where
a, b, c, d
∈
. The real part of
q
is
a
, and the purely imaginary part of
q
is
bi
+
cj
+
dk
.
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 Fall '08
 BROWNAWELL,WOODRO
 Algebra, Polynomials, Noncommutative Rings, division ring

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