Division algebra
An algebra
over a field
such that for any elements
and
the equations
 ,
are solvable in
 .
An associative division algebra, considered as a ring, is a skew-field, its centre
is a field, and
 .
If
 ,
the division algebra
is called a
central division algebra.
Finite-dimensional central associative division algebras over
may be identified, up to an isomorphism, with the elements of the
Brauer group
of the field
 .
Let
denote the dimension of
over
 .
If
and if
is the maximal subfield in
( ),
then
 .
According to the
Frobenius theorem,
all associative finite-dimensional division algebras over the field of real numbers
are exhausted by
itself, the field of complex numbers, and the
quaternion
algebra. For this reason the group
is cyclic of order two. If the associativity requirement is dropped, there is
yet another example of a division algebra over the field of real numbers: the
Cayley–Dickson algebra.
This algebra is alternative, and its dimension over
is 8. If
is a finite-dimensional (not necessarily associative) division algebra over
 ,
then
has one of the values 1, 2, 4, or 8.
References| [1] |
A.G. Kurosh,
"Lectures on general algebra"
, Chelsea
(1963)
(Translated from Russian) | | [2] |
A.A. Albert,
"Structure of algebras"
, Amer. Math. Soc.
(1939) | | [3] |
I.N. Herstein,
"Noncommutative rings"
, Math. Assoc. Amer.
(1968) | | [4] |
J.F. Adams,
"On the non-existence of elements of Hopf invariant one"
Ann. of Math.
, 72
: 1
(1960)
pp. 20–104 |
E.N. Kuz'min
CommentsOver a finite field every finite-dimensional central
division algebra is automatically commutative. For infinite-dimensional division algebras
the situation is quite different, because a result of
Mokar–Limonov states that such an algebra contains a free algebra in two variables.
If a finite-dimensional central division algebra
contains a maximal commutative subfield
which is a
Galois extension
of
 ,
then
is a
cross product
of
and
in the sense that
is the free
 -module
generated by
with product determined by:
Associativity of
entails that
represents an element of
(the second
Galois cohomology
group). One of the basic problems in algebra was formulated by
A. Albert
( 1931):
Is every finite-dimensional central division algebra necessarily a
cross product? In
1972,
S. Amitsur
provided a counter-example using
properties of generic division algebras resulting from the theory of PI-algebras (see
PI-algebra,
[a2]).
Other examples of division algebras were obtain by
F. van Ostaeyen
( 1972
Thesis, cf.
[a3]),
i.e. generic cross products, a notion generalized by
Amitsur and
D. Saltman
( 1978),
describing all cross
product division algebras for a given group
over the field
as reductions of a generic division algebra.
References| [a1] |
A.H. Schofield,
"Representations of rings over skew fields"
, London Math. Soc.
(1986) | | [a2] |
N. Jacobson,
"PI algebras. An introduction"
, Springer
(1975) | | [a3] |
F. van Oystaeyen,
"Prime spectra in non-commutative algebra"
, Springer
(1975) |
This text originally appeared in Encyclopaedia of Mathematics
- ISBN 1402006098
|