Division algebra

In the field ofmathematicscalledabstract algebra,adivision algebrais, roughly speaking, analgebra over a fieldin whichdivision,except by zero, is always possible.

Definitions

Formally, we start with anon-zero algebraDover afield.We callDadivision algebraif for any elementainDand any non-zero elementbinDthere exists precisely one elementxinDwitha=bxand precisely one elementyinDsuch thata=yb.

Forassociative algebras,the definition can be simplified as follows: a non-zero associative algebra over a field is adivision algebraif and only ifit has a multiplicativeidentity element1 and every non-zero elementahas a multiplicative inverse (i.e. an elementxwithax=xa= 1).

Associative division algebras

The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the fieldRofreal numbers,which are finite-dimensionalas avector spaceover the reals). TheFrobenius theoremstates thatup to isomorphismthere are three such algebras: the reals themselves (dimension 1), the field ofcomplex numbers(dimension 2), and thequaternions(dimension 4).

Wedderburn's little theoremstates that ifDis a finite division algebra, thenDis afinite field.^[1]

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

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

WheneverAis an associativeunital algebraover thefieldFandSis asimple moduleoverA,then theendomorphism ringofSis a division algebra overF;every associative division algebra overFarises in this fashion.

Thecenterof an associative division algebraDover the fieldKis 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 ofDover the center. Given a fieldF,theBrauer equivalenceclasses of simple (contains only trivial two-sided ideals) associative division algebras whose center isFand which are finite-dimensional overFcan be turned into a group, theBrauer groupof 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 algebrasandBanach algebras.

Not necessarily associative division algebras

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

Over the reals there are (up to isomorphism) only two unitarycommutativefinite-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 conjugateof the usual multiplication:

a*b={\overline {ab}}.

Thisis 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'stheorem, 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 algebrais 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 KervaireandJohn Milnorin 1958, again using techniques ofalgebraic topology,in particularK-theory.Adolf Hurwitzhad shown in 1898 that the identity $q{\overline {q}}={\text{sum of squares}}$ 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. Maysurveyed these attempts in 1966.^[4]

Any real finite-dimensional division algebra over the reals must be

isomorphic toRorCif unitary and commutative (equivalently: associative and commutative)
isomorphic to the quaternions if noncommutative but associative
isomorphic to theoctonionsif non-associative butalternative.

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

dimA= 1 ifKisalgebraically closed,
dimA= 1, 2, 4 or 8 ifKisreal closed,and
IfKis neither algebraically nor real closed, then there are infinitely many dimensions in which there exist division algebras overK.

We may say an algebraAhas multiplicative inversesif for any nonzero $a\in A$ there is an element $a^{-1}\in A$ with $aa^{-1}=a^{-1}a=1$ .An associative algebra has multiplicative inverses if and only if it is a division algebra. However, this fails for nonassociative algebras. Thesedenionsare 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 $i^{2}=-1+\epsilon j$ for some small nonzero real number $\epsilon$ while leaving the rest of the multiplication table unchanged. The element $i$ then has both right and left inverses, but they are not equal.