Page 479 - Advanced Linear Algebra
P. 479
An Introduction to Algebras 463
)
Note that 3 can be stated as follows: The product of two consecutive elements
)
)
Á Á is the next element with wraparound . Also, 4 says that &% ~ c%& for
(
%Á& ¸ Á Á ¹. This product is extended to all of by distributivity.
i
We leave it to the reader to verify that is a division algebra, called Hamilton's
i
)
quaternions, after their discoverer William Rowan Hamilton 1805-1865 .
(
(Readers familiar with group theory will recognize the quaternion group 8~
)
¸f Á f Á f Á f ¹. The quaternions have applications in geometry, computer
science and physics.
Finite-Dimensional Division Algebras over an Algebraically Closed
Field
It happens that there are no interesting finite-dimensional division algebras over
an algebraically closed field.
Theorem 18.10 If + is a finite-dimensional division algebra over an
algebraically closed field then + ~ - .
-
Proof. Let + have minimal polynomial ²%³ . Since a division algebra has
-
no zero divisors, ²%³ must be irreducible over and so must be linear.
Hence, ²%³ ~ % c and so ~ - .
Finite-Dimensional Division Algebras over a Finite Field
The finite-dimensional division algebras over a finite field are also easily
described: they are all commutative and so are finite fields. The proof, however,
is a bit more challenging. To understand the proof, we need two facts: the class
equation and some information about the complex roots of unity. So let us
briefly describe what we need.
The Class Equation
Those who have studied group theory have no doubt encountered the famous
.
class equation. Let be a finite group. Each . can be thought of as a
permutation of defined by
.
%~ % c
%
for all % . . The set of all conjugates % c of is denoted by ¸%¹ . and so
.
¸%¹ ~ ¸ % .¹
This set is also called a conjugacy class in . . Now, the following are
equivalent:
