[permalink] [id link]

* Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra ; in fact, it is either a two-by-two matrix algebra, or a division algebra.

from
Wikipedia

## Some Related Sentences

Every and central

__Every__telephone company

**,**whether large

**or**small

**,**determines its own ANAC for each individual

__central__office

**,**which tends

**to**perpetuate the current situation of

**a**mess of overlapping and /

**or**spotty areas of coverage

**.**

__Every__year the

__central__business district ( with corners at the Municipal Building

**,**Grand Street Fire House and Croton-Harmon High School )

**is**closed

**to**automobile traffic for music

**,**American food

**,**local fund raisers

**,**traveling

**,**and local artists

**.**

*****

__Every__finite-dimensional

**simple**

**algebra**

**over**R must be

**a**

**matrix**ring

**over**R

**,**C

**,**

**or**H

**.**

__Every__

__central__

**simple**

**algebra**

**over**R must be

**a**

**matrix**ring

**over**R

**or**H

**.**These results follow from the Frobenius theorem

**.**

*****

__Every__finite-dimensional

**simple**

**algebra**

**over**C must be

**a**

**matrix**ring

**over**C and hence every

__central__

**simple**

**algebra**

**over**C must be

**a**

**matrix**ring

**over**C

**.**

*****

__Every__finite-dimensional

__central__

**simple**

**algebra**

**over**

**a**finite

**field**must be

**a**

**matrix**ring

**over**that

**field**

**.**

*****

__Every__automorphism of

**a**

__central__

**simple**

**algebra**

**is**an inner automorphism ( follows from Skolem – Noether theorem ).

Dual citizenship was established ( Article 5 ( 3 ): "

__Every__Czechoslovak citizen**is**at the same time**a**citizen of the Czech Socialist Republic**or**the Slovak Socialist Republic "), and many of the former functions of the__central__government were instead placed under the jurisdiction of the two national governments**.**__Every__telephone company

**,**whether large

**or**small

**,**determines its own ringback numbers for each individual

__central__office

**,**which tends

**to**perpetuate the current situation of

**a**mess of overlapping and /

**or**spotty areas of coverage

**.**

__Every__summer

**,**

**a**tourist train connects the ancient parts of the city with museums

**,**the

__central__railway station and the passenger ship terminal of Krems

**.**

__Every__year

**,**at the end of January

**,**the offerings of the members of the society all

**over**the world are forwarded

**to**these

__central__bureaux

**,**and the total amount

**is**divided among all the missions of the earth

**.**

*****Retzai

__Every__the Orthodox Amidah prayer

**,**the

__central__prayer of Jewish services

**,**contains the paragraph: " Be favorable

**,**Oh Lord our God

**,**

**to**your people Israel and their prayer

**,**and restore the service of the Holy of Holies of Your House

**,**and accept the fire-offerings of Israel and their prayer with love and favor

**,**and may the service of your people Israel always be favored

**.**

*****County Party:

__Every__two years

**,**between the primary election and 2 weeks after the general election

**,**all the precinct committee people-the county party "

__central__Committee "-hold

**a**meeting and elect officers for the county party

**,**

**a**chair

**,**vice chair

**,**secretary and treasurer

**.**

__Every__connected topological group has

**a**unique universal cover as

**a**topological space

**,**which has

**a**unique group structure as

**a**

__central__extension by the fundamental group

**.**

__Every__year

**,**thousands of

__central__police and state police personnel are deployed

**to**provide security

**to**pilgrims from potential terror threats

**.**

__Every__glucosinolate contains

**a**

__central__carbon atom

**,**which

**is**bound via

**a**sulfur atom

**to**the thioglucose group ( making

**a**sulfated ketoxime ) and via

**a**nitrogen atom

**to**

**a**sulfate group

**.**

In mathematics

**,****a****quaternion****algebra****over****a****field****F****is****a**__central__**simple****algebra**A**over****F**that has dimension 4**over****F****.**__Every__**quaternion****algebra**becomes the**matrix****algebra**by extending scalars (= tensoring with**a****field**extension ), i**.**e**.**for**a**suitable**field**extension K of**F****,****is****isomorphic****to**the 2 × 2**matrix****algebra****over**K**.**

Every and simple

#

__Every____simple__path from**a**given node**to**any of its descendant leaves contains the same number of black nodes**.**__Every__

__simple__R-module

**is**

**isomorphic**

**to**

**a**quotient R / m where m

**is**

**a**maximal right ideal of R

**.**By the above paragraph

**,**any quotient R / m

**is**

**a**

__simple__module

**.**

" I might easily have written this story

**in**the traditional manner [...]__Every__novelist knows the recipe [...] It**is**not very difficult**to**follow**a**__simple__**,**chronological scheme which the critics will understand [...] But I**,**after all**,**am trying**to**tell the story of this Chapelizod family**in****a**new way**.**__Every__October

**,**Moriarty plays host

**to**the Pinto Bean Fiesta

**,**which

**is**composed of

**a**bunch of

__simple__games

**in**Crossly Park

**,**as well as

**a**parade and crowning of

**a**" Pinto Bean Queen

**.**

__Every__one of the infinitely many vertices of G can be reached from v < sub > 1 </ sub > with

**a**

__simple__path

**,**and each such path must start with one of the finitely many vertices adjacent

**to**v < sub > 1 </ sub >.

__Every__hour that Napoleon could have attacked earlier as he did

**,**would have been

**is**his favour

**,**but the French could not attack

**in**the morning for the

__simple__reason that the entire army had not yet taken its battle positions

**.**

__Every__closed curve c on X

**is**homologous

**to**for some

__simple__closed curves c < sub > i </ sub >, that

**is**

**,**

# Personal right:

__Every__person has**a**right**to**life but this right**is**restricted and has attached certain duties –__simple__living**is**essential**.**__Every__Sámi settlement had its seita

**,**which had no regular shape

**,**and might consist of smooth

**or**odd-looking stones picked out of

**a**stream

**,**of

**a**small pile of stones

**,**of

**a**tree-stump

**,**

**or**of

**a**

__simple__post

**.**

Every and algebra

__Every__associative

__algebra__

**is**obviously alternative

**,**but so too are some strictly nonassociative algebras such as the octonions

**.**

__Every__Boolean

__algebra__( A

**,**∧, ∨) gives rise

**to**

**a**ring ( A

**,**+, ·) by defining

**a**+ b := (

**a**∧ ¬ b ) ∨ ( b ∧ ¬

**a**) = (

**a**∨ b ) ∧ ¬(

**a**∧ b ) ( this operation

**is**called symmetric difference

**in**the case of sets and XOR

**in**the case of logic ) and

**a**· b :=

**a**∧ b

**.**The zero element of this ring coincides with the 0 of the Boolean

__algebra__

**;**the multiplicative identity element of the ring

**is**the 1 of the Boolean

__algebra__

**.**

*****

__Every__real Banach

__algebra__which

**is**

**a**

**division**

__algebra__

**is**

**isomorphic**

**to**the reals

**,**the complexes

**,**

**or**the quaternions

**.**

*****

__Every__unital real Banach

__algebra__with no zero divisors

**,**and

**in**which every principal ideal

**is**closed

**,**

**is**

**isomorphic**

**to**the reals

**,**the complexes

**,**

**or**the quaternions

**.**

*****

__Every__commutative real unital Noetherian Banach

__algebra__with no zero divisors

**is**

**isomorphic**

**to**the real

**or**complex numbers

**.**

*****

__Every__commutative real unital Noetherian Banach

__algebra__( possibly having zero divisors )

**is**finite-dimensional

**.**

__Every__continuous map f: X → Y induces an

__algebra__homomorphism C ( f ): C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f for every φ

**in**C ( Y ).

__Every__vector v

**in**determines

**a**linear map from R

**to**taking 1

**to**v

**,**which can be thought of as

**a**Lie

__algebra__homomorphism

**.**

__Every__associative

__algebra__

**is**obviously power-associative

**,**but so are all other alternative algebras ( like the octonions

**,**which are non-associative ) and even some non-alternative algebras like the sedenions

**.**

__Every__random vector gives rise

**to**

**a**probability measure on R < sup > n </ sup > with the Borel

__algebra__as the underlying sigma-algebra

**.**

__Every__finite-dimensional Hausdorff topological vector space

**is**reflexive

**,**because J

**is**bijective by linear

__algebra__

**,**and because there

**is**

**a**unique Hausdorff vector space topology on

**a**finite dimensional vector space

**.**

__Every__Boolean

__algebra__can be obtained

**in**this way from

**a**suitable topological space: see Stone's representation theorem for Boolean algebras

**.**

__Every__Boolean

__algebra__

**is**

**a**Heyting

__algebra__when

**a**→ b

**is**defined as usual as ¬

**a**∨ b

**,**as

**is**every complete distributive lattice when

**a**→ b

**is**taken

**to**be the supremum of the set of all c for which

**a**∧ c ≤ b

**.**The open sets of

**a**topological space form

**a**complete distributive lattice and hence

**a**Heyting

__algebra__

**.**

__Every__Heyting

__algebra__with exactly one coatom

**is**subdirectly irreducible

**,**whence every Heyting

__algebra__can be made an SI by adjoining

**a**new top

**.**

0.476 seconds.