In the field ofmathematics calledabstract algebra, adivision algebra is, roughly speaking, analgebra over a field in whichdivision, except by zero, is always possible.

Formally, we start with anon-zeroalgebraD over afield. We callD adivision algebra if for any elementa inD and any non-zero elementb inD there exists precisely one elementx inD witha =bx and precisely one elementy inD such thata =yb.

Forassociative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is adivision algebraif and only if it has a multiplicativeidentity element 1 and every non-zero elementa has a multiplicative inverse (i.e. an elementx withax =xa = 1).

The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the fieldR ofreal numbers, which are finite-dimensional as avector space over the reals). TheFrobenius theorem states thatup toisomorphism there are three such algebras: the reals themselves (dimension 1), the field ofcomplex numbers (dimension 2), and thequaternions (dimension 4).

Wedderburn's little theorem states that ifD is a finite division algebra, thenD is afinite field.^[1]

Over analgebraically closed fieldK (for example thecomplex numbersC), there are no finite-dimensional associative division algebras, exceptK itself.^[2]

Associative division algebras have no nonzerozero divisors. Afinite-dimensionalunitalassociative algebra (over any field) is a division algebraif and only if it has no nonzero zero divisors.

WheneverA is an associativeunital algebra over thefieldF andS is asimple module overA, then theendomorphism ring ofS is a division algebra overF; every associative division algebra overF arises in this fashion.

Thecenter of an associative division algebraD over the fieldK is a field containingK. The dimension of such an algebra over its center, if finite, is aperfect square: it is equal to the square of the dimension of a maximal subfield ofD over the center. Given a fieldF, theBrauer equivalence classes of simple (contains only trivial two-sided ideals) associative division algebras whose center isF and which are finite-dimensional overF can be turned into a group, theBrauer group of the fieldF.

One way to construct finite-dimensional associative division algebras over arbitrary fields is given by thequaternion algebras (see alsoquaternions).

For infinite-dimensional associative division algebras, the most important cases are those where the space has some reasonabletopology. See for examplenormed division algebras andBanach algebras.

If the division algebra is not assumed to be associative, usually some weaker condition (such asalternativity orpower associativity) is imposed instead. Seealgebra over a field for a list of such conditions.

Over the reals there are (up to isomorphism) only two unitarycommutative finite-dimensional division algebras: the reals themselves, and the complex numbers. These are of course both associative. For a non-associative example, consider the complex numbers with multiplication defined by taking thecomplex conjugate of the usual multiplication:

${\displaystyle a\ast b=\overline{ab}.}$

This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element. There are infinitely many other non-isomorphic commutative, non-associative, finite-dimensional real divisional algebras, but they all have dimension 2.

In fact, every finite-dimensional real commutative division algebra is either 1- or 2-dimensional. This is known asHopf's theorem, and was proved in 1940. The proof uses methods fromtopology. Although a later proof was found usingalgebraic geometry, no direct algebraic proof is known. Thefundamental theorem of algebra is a corollary of Hopf's theorem.

Dropping the requirement of commutativity, Hopf generalized his result: Any finite-dimensional real division algebra must have dimension a power of 2.

Later work showed that in fact, any finite-dimensional real division algebra must be of dimension 1, 2, 4, or 8. This was independently proved byMichel Kervaire andJohn Milnor in 1958, again using techniques ofalgebraic topology, in particularK-theory.Adolf Hurwitz had shown in 1898 that the identity${\displaystyle q\overline{q}=\text{sum of squares}}$ held only for dimensions 1, 2, 4 and 8.^[3] (SeeHurwitz's theorem.) The challenge of constructing a division algebra of three dimensions was tackled by several early mathematicians.Kenneth O. May surveyed these attempts in 1966.^[4]

Any real finite-dimensional division algebraover the reals must be

• isomorphic toR orC if unitary and commutative (equivalently: associative and commutative)
• isomorphic to the quaternions if noncommutative but associative
• isomorphic to theoctonions if non-associative butalternative.

The following is known about the dimension of a finite-dimensional division algebraA over a fieldK:

• dimA = 1 ifK isalgebraically closed,
• dimA = 1, 2, 4 or 8 ifK isreal closed, and
• IfK is neither algebraically nor real closed, then there are infinitely many dimensions in which there exist division algebras overK.

We may say an algebraAhas multiplicative inverses if for any nonzero${\displaystyle a\in A}$ there is an element${\displaystyle a^{-1}\in A}$ with${\displaystyle a{a}^{-1}=a^{-1}a=1}$. An associative algebra has multiplicative inverses if and only if it is a division algebra. However, this fails for nonassociative algebras. Thesedenions are a nonassociative algebra over the real numbers that has multiplicative inverses, but is not a division algebra. On the other hand, we can construct a division algebra without multiplicative inverses by taking the quaternions and modifying the product, setting${\displaystyle i^2=-1+ϵj}$ for some small nonzero real number${\displaystyle ϵ}$ while leaving the rest of the multiplication table unchanged. The element${\displaystyle i}$ then has both right and left inverses, but they are not equal.

• Normed division algebra
• Division (mathematics)
• Division ring
• Semifield
• Cayley–Dickson construction

• Cohn, Paul Moritz (2003).Basic algebra: groups, rings, and fields. London: Springer-Verlag.doi:10.1007/978-0-85729-428-9.ISBN 978-1-85233-587-8.MR 1935285.
• Lam, Tsit-Yuen (2001).A first course in noncommutative rings.Graduate Texts in Mathematics. Vol. 131 (2 ed.). Springer.ISBN 0-387-95183-0.

• "Division algebra",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Algebras
• Ring theory

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