Difference between revisions of "Division algebra"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
||
Line 2: | Line 2: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) {{MR|0158000}} {{ZBL|0121.25901}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) {{MR|0000595}} {{ZBL|0023.19901}} {{ZBL|65.0094.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) {{MR|1535024}} {{MR|0227205}} {{ZBL|0177.05801}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.F. Adams, "On the non-existence of elements of Hopf invariant one" ''Ann. of Math.'' , '''72''' : 1 (1960) pp. 20–104 {{MR|0141119}} {{ZBL|0096.17404}} </TD></TR></table> |
Line 16: | Line 16: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.H. Schofield, "Representations of rings over skew fields" , London Math. Soc. (1986) {{MR|0800853}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Jacobson, "PI algebras. An introduction" , Springer (1975) {{MR|0369421}} {{ZBL|0326.16013}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> F. van Oystaeyen, "Prime spectra in non-commutative algebra" , Springer (1975) {{MR|}} {{ZBL|0302.16001}} </TD></TR></table> |
Revision as of 17:32, 31 March 2012
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) MR0158000 Zbl 0121.25901 |
[2] | A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) MR0000595 Zbl 0023.19901 Zbl 65.0094.02 |
[3] | I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) MR1535024 MR0227205 Zbl 0177.05801 |
[4] | J.F. Adams, "On the non-existence of elements of Hopf invariant one" Ann. of Math. , 72 : 1 (1960) pp. 20–104 MR0141119 Zbl 0096.17404 |
Comments
Over 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:
![]() | (a1) |
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) MR0800853 |
[a2] | N. Jacobson, "PI algebras. An introduction" , Springer (1975) MR0369421 Zbl 0326.16013 |
[a3] | F. van Oystaeyen, "Prime spectra in non-commutative algebra" , Springer (1975) Zbl 0302.16001 |
Division algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Division_algebra&oldid=24066