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| In the field of [[mathematics]] called [[abstract algebra]], a '''division algebra''' is, roughly speaking, an [[algebra over a field]], in which [[division (mathematics)|division]] is possible.
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| == Definitions ==
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| Formally, we start with an [[Algebra over a field|algebra]] ''D'' over a [[field (mathematics)|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''.
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| For [[associative algebra]]s, 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).
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| ==Associative division algebras==
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| The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field '''R''' of [[real number]]s, which are finite-[[Hamel dimension|dimensional]] as a [[vector space]] over the reals). The [[Frobenius theorem (real division algebras)|Frobenius theorem]] states that [[up to]] [[isomorphism]] there are three such algebras: the reals themselves (dimension 1), the field of [[complex number]]s (dimension 2), and the [[quaternions]] (dimension 4). | |
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| [[Wedderburn's little theorem]] states that if ''D'' is a finite division algebra, then ''D'' is a [[finite field]].<ref>Lam (2001), {{Google books quote|id=f15FyZuZ3-4C|page=203|text=Wedderburn's "little" theorem|p. 203}}</ref>
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| Over an [[algebraically closed field]] ''K'' (for example the [[complex number]]s '''C'''), there are no finite-dimensional associative division algebras, except ''K'' itself.<ref>Cohn (2003), {{Google books quote|id=VESm0MJOiDQC|page=150|text=only division algebra|Proposition 5.4.5, p. 150}}</ref>
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| Associative division algebras have no [[zero divisor]]s. A ''finite-dimensional'' [[unital algebra|unital]] [[associative algebra]] (over any field) is a division algebra ''if and only if'' it has no zero divisors.
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| Whenever ''A'' is an associative [[unital algebra]] over the [[field (mathematics)|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.
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| The [[center (algebra)|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 [[square number|perfect square]]: it is equal to the square of the dimension of a maximal subfield of ''D'' over the center. Given a field ''F'', the [[Brauer equivalence]] classes of simple (contains only trivial two-sided ideals) associative division algebras whose center is ''F'' and which are finite-dimensional over ''F'' can be turned into a group, the [[Brauer group]] of the field ''F''.
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| One way to construct finite-dimensional associative division algebras over arbitrary fields is given by the [[quaternion algebra]]s (see also [[quaternion]]s).
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| For infinite-dimensional associative division algebras, the most important cases are those where the space has some reasonable [[topology]]. See for example [[normed division algebra]]s and [[Banach algebra]]s.
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| ==Not necessarily associative division algebras==<!-- This section is linked from [[Division ring]] -->
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| If the division algebra is not assumed to be associative, usually some weaker condition (such as [[alternativity]] or [[power associativity]]) is imposed instead. See [[algebra over a field]] for a list of such conditions.
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| Over the reals there are (up to isomorphism) only two unitary [[commutative]] 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 the [[complex conjugate]] of the usual multiplication:
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| :<math>a*b=\overline{ab}.</math>
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| [[Example of a non-associative algebra|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.
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| In fact, every finite-dimensional real commutative division algebra is either 1 or 2 dimensional. This is known as [[Heinz Hopf|Hopf's]] theorem, and was proved in 1940. The proof uses methods from [[topology]]. Although a later proof was found using [[algebraic geometry]], no direct algebraic proof is known. The [[fundamental theorem of algebra]] is a corollary of Hopf's theorem.
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| Dropping the requirement of commutativity, Hopf generalized his result: Any finite-dimensional real division algebra must have dimension a power of 2.
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| 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 by [[Michel Kervaire]] and [[John Milnor]] in 1958, again using techniques of [[algebraic topology]], in particular [[K-theory]]. [[Adolf Hurwitz]] had shown in 1898 that the identity <math>q\overline{q} = \textrm{sum\ of\ squares}</math> held only for dimensions 1, 2, 4 and 8.<ref>{{cite book|title=[[The Road To Reality]]|authorlink=Roger Penrose|author=Roger Penrose|year=2005|publisher=Vintage|isbn=0-09-944068-7}}</ref> (See [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]].)
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| While there are infinitely many non-isomorphic real division algebras of dimensions 2, 4 and 8, one can say the following: any real finite-dimensional division algebra
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| over the reals must be
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| * isomorphic to '''R''' or '''C''' if unitary and commutative (equivalently: associative and commutative)
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| * isomorphic to the quaternions if noncommutative but associative
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| * isomorphic to the [[octonions]] if non-associative but [[alternative algebra|alternative]].
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| The following is known about the dimension of a finite-dimensional division algebra ''A'' over a field ''K'':
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| * dim ''A'' = 1 if ''K'' is [[algebraically closed]],
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| * dim ''A'' = 1, 2, 4 or 8 if ''K'' is [[real closed]], and
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| * If ''K'' is neither algebraically nor real closed, then there are infinitely many dimensions in which there exist division algebras over ''K''.
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| ==See also==
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| * [[Normed division algebra]]
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| * [[Division (mathematics)]]
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| * [[Division ring]]
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| * [[Semifield]]
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| ==Notes==
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| <references/>
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| == References ==
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| * {{cite book |last1=Cohn |first1=Paul Moritz |authorlink1= |last2= |first2= |authorlink2= |title=Basic algebra: groups, rings, and fields |url= |edition= |series= |volume= |year=2003 |publisher=Springer |location= |isbn=978-1-85233-587-8 |id= }}
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| * {{cite book |last1=Lam |first1=Tsit-Yuen |authorlink1= |last2= |first2= |authorlink2= |title=A first course in noncommutative rings |url= |edition=2 |series=Graduate texts in mathematics |volume=131 |year=2001 |publisher=Springer |location= |isbn=0-387-95183-0 |id= }}
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| ==External links==
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| * {{springer|title=Division algebra|id=p/d033680}}
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| [[Category:Algebras]]
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| [[Category:Ring theory]]
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With the backdrop of the Superstition Mountains, you have a scenic mountain bike trail network that can embody every quality of off-road riding in the Southwest. The mountain bike park is within easy riding distance, approximately 3km from the town centre. High end mountain bikes use carbon fibre frames, or other, more exotic materials to reduce weight and keep stiffness up. Generally speaking, a good, durable mountain bike starts at about $600 and goes up from there. As the area has grown, affordable housing needs have to.
You can choose from many different suspensions on your bike, make sure the suspension you choose is going to fit the type of cycling you intend for it. Now most people wear gloves whilst they are riding because they keep the hands warm. You can spend under $100 for a bargain bike at a department store, or lay down thousands for a professional model. Many riders suggest that if you have less then $700 to spend on a bike you might want to consider a hardtail because full suspension bikes are more costly. It will be business as normal, very successful business as normal.
They are electricity assisted, meaning electric power is used in response to the rider's pedaling power, with pedaling staying the key driver. That's a level of quality you just don't get from many brands these days. Buy a bike that has a top quality body made outside of steel, aluminum, carbon fiber or titanium. In no time you'll be cruising through the woods and getting into great shape. Re-lube the chain carefully, making sure each link is covered with oil, spin the pedals backwards a few times, then wipe off any excess oil from the chain with an old rag.
Decide how and where you want to ride, then choose a mountain bike that fits your personal style, rather than that of the sales staff. Find more information about Hardtail mountain bikes here. It is a quality piece available at affordable price. Should you have virtually any concerns regarding where and tips on how to work with Popular mountain bike sizing., you are able to e mail us from our web site. Instead of having fun playing with him (like a child with a favorite playmate), they evaluate his potential as a father. 7,934,739) CVA is an award-winning cancellation style and now it is recognized with a US Apparent.
And there are some people who rightfully bring up the issue of the fuel costs and environmental impact of producing the electricty used to power the bike in the first place. If you can, try and drop off a gutter or curb, standing to allow your knees to absorb the impact. I hope what you will have found in the article with respect to Mongoose mountain bike reviews, together with additionally the particular info regarding mountain bikes, is going to be helpful for you. A twenty-inch wheeled bike are good for longer traveling time. In some countries, they still have the culture of using the bike as their primary vehicles used on their streets.