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) 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