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	The ~~writer~~'s ~~title is Christy Brookins~~. ~~To perform lacross~~ is the ~~factor I love psychic readings~~ [[~~http~~:/~~/mybrandcp~~.~~com/xe/board_XmDx25/107997 supplemental resources~~]] ~~most~~ of ~~all~~. I'~~ve always cherished living~~ in ~~Kentucky but now I~~'~~m considering other choices~~. He is ~~an order clerk and it~~'s ~~something he really appreciate~~.		In [[mathematics]], an '''octonion algebra''' or '''Cayley algebra''' over a [[field (mathematics)\|field]] ''F'' is an [[algebraic structure]] which is an 8-[[Dimension (vector space)\|dimensional]] [[composition algebra]] over ''F''. In other words, it is a [[unital algebra\|unital]] [[Algebra over a field#Non-associative algebras\|nonassociative algebra]] ''A'' over ''F'' with a [[Degeneracy (mathematics)\|nondegenerate]] [[quadratic form]] ''N'' (called the ''norm form'') such that
			:<math>N(xy) = N(x)N(y)</math>
			for all ''x'' and ''y'' in ''A''.

			The most well-known example of an octonion algebra are the classical [[octonion]]s, which are an octonion algebra over '''R''', the field of [[real number]]s. The [[split-octonion]]s also form an octonion algebra over '''R'''. Up to [[Algebra homomorphism\|'''R'''-algebra isomorphism]], these are the only octonion algebras over the reals.

			The octonion algebra for ''N'' is a [[division algebra]] if and only if the form ''N'' is [[Anisotropic quadratic space\|anisotropic]]: a '''split octonion algebra''' is one for which the quadratic form ''N'' is [[Isotropic quadratic form\|isotropic]] (i.e. there exists a non-zero vector ''x'' with ''N''(''x'') = 0). Up to ''F''-algebra isomorphism, there is a unique split octonion algebra over any field ''F''.<ref name=Sch48>Schafer (1995) p.48</ref> When ''F'' is [[algebraically closed field\|algebraically closed]] or a [[finite field]], these are the only octonion algebras over ''F''.

			Octonion algebras are always nonassociative. They are however [[alternative algebra]]s (a weaker form of associativity). Moreover, the [[Moufang identities]] hold in any octonion algebra. It follows that the set of invertible elements in any octonion algebra form a [[Moufang loop]], as do the subset of unit norm elements.

			== Classification ==

			It is a theorem of [[Adolf Hurwitz]] that the ''F''-isomorphism classes of the norm form are in one-to-one correspondence with the isomorphism classes of octonion ''F''-algebras. Moreover, the possible norm forms are exactly the [[Pfister form\|Pfister 3-forms]] over ''F''.<ref name=Lam327>Lam (2005) p.327</ref>

			Since any two octonion ''F''-algebras become isomorphic over the algebraic closure of ''F'', one can apply the ideas of non-abelian [[Galois cohomology]]. In particular, by using the fact that the automorphism group of the split octonions is the split [[algebraic group]] [[G2 (mathematics)\|G<sub>2</sub>]], one sees the correspondence of isomorphism classes of octonion ''F''-algebras with isomorphism classes of G<sub>2</sub>-[[torsor]]s over ''F''. These isomorphism classes form the non-abelian Galois cohomology set <math>H^1(F, G_2)</math>.<ref name=GMS>Garibaldi, Merkurjev & Serre (2003) pp.9-10,44</ref>

			==See also==

			*[[quaternion algebra]]

			==References==
			{{reflist}}
			* {{cite book \| last1=Garibaldi \| first1=Skip \| author1-link=Skip Garibaldi \| last2=Merkurjev \| first2=Alexander \| author2-link=Alexander Merkurjev \| last3=Serre \| first3=Jean-Pierre \| author3-link=Jean-Pierre Serre \| title=Cohomological invariants in Galois cohomology \| series=University Lecture Series \| volume=28 \| location=Providence, RI \| publisher=[[American Mathematical Society]] \| year=2003 \| isbn=0-8218-3287-5 \| zbl=1159.12311 }}
			* {{cite book \| title=Introduction to Quadratic Forms over Fields \| volume=67 \| series=Graduate Studies in Mathematics \| first=Tsit-Yuen \| last=Lam \| authorlink=Tsit Yuen Lam \| publisher=[[American Mathematical Society]] \| year=2005 \| isbn=0-8218-1095-2 \| zbl=1068.11023 \| mr = 2104929 }}
			* {{cite book \| first=Richard D. \| last=Schafer \| year=1995 \| origyear=1966 \| zbl=0145.25601 \| title=An introduction to non-associative algebras \| publisher=[[Dover Publications]] \| isbn=0-486-68813-5 }}
			*{{cite book \| first = J. P. \| last = Serre \| authorlink = Jean-Pierre Serre \| year = 2002 \|title = Galois Cohomology \| publisher = [[Springer-Verlag]] \| zbl= 1004.12003 \| others=Translated from the French by Patrick Ion \| series=Springer Monographs in Mathematics \| location=Berlin \| isbn=3-540-42192-0 }}
			*{{cite book
			\| first1 = T. A.
			\| last1 = Springer
			\| author1-link = T. A. Springer
			\| first2 = F. D.
			\| last2 = Veldkamp
			\| year = 2000
			\| title = Octonions, Jordan Algebras and Exceptional Groups
			\| publisher = Springer-Verlag
			\| id = ISBN 3-540-66337-1
			}}

			==External links==
			* {{SpringerEOM \| title=Cayley–Dickson algebra \| id= 22269 }}

			[[Category:Octonions\|Algebra]]
			[[Category:Non-associative algebras]]

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