Division algebra | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Formally, we start with a non-zero algebra D over a field. 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: a non-zero associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1 and every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1). 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. Over an algebraically closed field K (for example the complex numbers C), there are no finite-dimensional associative division algebras, except K itself. Associative division algebras have no nonzero zero divisors. A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no nonzero 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. The center of an associative division algebra D over the field K is a field containing K. The dimension of such an algebra over its center, if finite, is a perfect square: it is equal to the square of the dimension of a maximal subfield of D over the center.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (1)

Related concepts (44)

Related courses (6)

Related lectures (40)

Noncommutative ring

In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings. Sometimes the term noncommutative ring is used instead of ring to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative.

Hurwitz's theorem (composition algebras)

In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions.

Division algebra

MATH-334: Representation theory

Study the basics of representation theory of groups and associative algebras.

MATH-432: Probability theory

The course is based on Durrett's text book Probability: Theory and Examples.
It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.

MATH-215: Rings and fields

C'est un cours introductoire dans la théorie d'anneau et de corps.

Lattice theoretic approach to space-time codes from division algebras

Cyril Becker

This master project on algebraic coding theory gathers various techniques from lattice theory, central simple algebras and algebraic number theory. The thesis begins with the formulation of the engineering problem into mathematical form. It presents how space-time codes appeared, and how we can construct codes based on Q(i)-central division algebras. More precisely, it is explained how we can build codes from cyclic division algebras; and a generalization is done with the introduction of crossed product algebras. The approach used is slightly more geometric than what is usually done in the literature. We also give slight generalization to known methods for analyzing minimum determinants of division algebra-based codes.

2012

Explores Wedderburn's theorem, group algebras, and Maschke's theorem in the context of finite dimensional simple algebras and their endomorphisms.

Covers the goals and motivations of representation theory, focusing on associative algebras and homomorphisms.

Explores convolution, uniform continuity, Hilbert structure, and Lebesgue measure in analysis.