Division algebra

related topics
{math, number, function}

In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.

Contents

Definitions

Formally, we start with an algebra D over a field, and assume that D does not just consist of its zero element. We call D a division algebra if for any element a in D and any non-zero element b in D there exists precisely one element x in D with a = bx and precisely one element y in D such that a = yb.

For associative algebras, the definition can be simplified as follows: an associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1≠0 and every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1).

Associative division algebras

The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite-dimensional as a vector space over the reals). The Frobenius theorem states that up to isomorphism there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4).

Wedderburn's little theorem states that if D is a finite division algebra, then D is a finite field. (T. Y. Lam, A First Course in Noncommutative Rings.)

Over an algebraically closed field K (for example the complex numbers C), there are no finite-dimensional associative division algebras, except K itself of course.

Associative division algebras have no zero divisors. A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no zero divisors.

Whenever A is an associative unital algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every associative division algebra over F arises in this fashion.

Full article ▸

related documents
Exact sequence
Search algorithm
Separable space
Mersenne prime
Golomb coding
Probability space
Symmetric group
Banach fixed point theorem
Euler's totient function
Linear equation
Kruskal's algorithm
Mersenne twister
Tychonoff space
Local ring
Separation axiom
Controllability
Gamma function
Topological group
Group representation
Goodstein's theorem
Well-order
Diophantine set
Lebesgue measure
Quine (computing)
Heapsort
Kolmogorov space
Polytope
Laurent series
Julia set
Blackboard bold