Elementary number theory

An algebra $ A $ over a field $ F $ such that for any elements $ a \neq 0 $ and $ b $ the equations $ ax = b $, $ ya = b $ are solvable in $ A $. An associative division algebra, considered as a ring, is a skew-field, its centre $ C $ is a field, and $ C \supseteq F $. If $ C = F $, the division algebra $ A $ is called a central division algebra. Finite-dimensional central associative division algebras over $ F $ may be identified, up to an isomorphism, with the elements of the Brauer group $ B( F ) $ of the field $ F $. Let $ [ A: F ] $ denote the dimension of $ A $ over $ F $. If $ A \in B( F ) $ and if $ L $ is the maximal subfield in $ A $ ($ L \supseteq F $), then $ [ A: F ] = {[ L: F ] } ^ {2} $. According to the Frobenius theorem, all associative finite-dimensional division algebras over the field of real numbers $ \mathbf R $ are exhausted by $ \mathbf R $ itself, the field of complex numbers, and the quaternion algebra. For this reason the group $ B( \mathbf R ) $ 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 $ \mathbf R $ is 8. If $ A $ is a finite-dimensional (not necessarily associative) division algebra over $ \mathbf R $, then $ [ A: \mathbf R ] $ 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 $ D $ contains a maximal commutative subfield $ L $ which is a Galois extension of $ F $, then $ D $ is a cross product of $ L $ and $ G = \mathop{\rm Gal} ( L/ F ) $ in the sense that $ D $ is the free $ L $-module generated by $ \{ {u _ \sigma } : {\sigma \in G } \} $ with product determined by:

$$ \tag{a1 } \left . \begin{array}{ll} u _ \sigma u _ \tau = c ( \sigma , \tau ) u _ {\sigma \tau } &\textrm{ for some } c ( \sigma , \tau ) \in L ^ {*} , \\ u _ \sigma \lambda = \lambda ^ \sigma u _ \sigma &\textrm{ for } \lambda \in L ,\ \tau \in G . \\ \end{array} \right \} $$

Associativity of $ D $ entails that $ c : G \times G \rightarrow L ^ {*} $ represents an element of $ H ^ {2} ( G , L ^ {*} ) $ (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 $ G $ over the field $ F $ 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