Content deleted Content added
Toploftical (talk | contribs) link |
|||
(45 intermediate revisions by 20 users not shown) | |||
Line 1:
{{Short description|Generalization of quaternions to other fields}}
In [[mathematics]], a '''quaternion algebra''' over a [[field (mathematics)|field]] ''F'' is a [[central simple algebra]] ''A'' over ''F''<ref>See
The notion of a quaternion algebra can be seen as a generalization of
==Structure==
''Quaternion algebra'' here means something more general than the [[algebra over a field|algebra]] of
:<math>i^2=a</math>
:<math>j^2=b</math>
:<math>ij=k</math>
:<math>ji=-k</math>
where ''a'' and ''b'' are any given nonzero elements of ''F''. From these rules we get:
:<math>k^2=ijij=-iijj=-ab</math>
The classical instances where <math>F=\mathbb{R}</math> are Hamilton's quaternions (''a'' = ''b'' = −1) and [[split-quaternion]]s (''a'' = −1, ''b'' = +1). In split-quaternions, <math>k^2 = +1</math> and <math>j k = - i </math>, differing from Hamilton's equations.
(The Hamilton quaternions are the case where <math>F=\mathbb{R}</math> and ''a'' = ''b'' = −1.) The algebra defined in this way is denoted (''a'',''b'')<sub>''F''</sub> or simply (''a'',''b'').<ref name=GS2>Gille & Szamuely (2006) p.2</ref> When ''F'' has characteristic 2, a different explicit description in terms of a basis of 4 elements is also possible, but in any event the definition of a quaternion algebra over ''F'' as a 4-dimensional central simple algebra over ''F'' applies uniformly in all characteristics.▼
▲
A quaternion algebra (''a'',''b'')<sub>''F''</sub> is either a [[division algebra]] or isomorphic to the [[matrix algebra]] of 2×2 matrices over ''F'': the latter case is termed ''split''.<ref name=GS3>Gille & Szamuely (2006) p.3</ref> The ''norm form''▼
▲A quaternion algebra (''a'',''b'')<sub>''F''</sub> is either a [[division algebra]] or isomorphic to the [[matrix algebra]] of
:<math>N(t + xi +yj + zk) = t^2 - ax^2 - by^2 + abz^2 \ </math>
defines a structure of [[division algebra]] if and only if the norm is an [[anisotropic quadratic form]], that is, zero only on the zero element. The [[conic section|conic]] ''C''(''a'',''b'') defined by
:<math>a x^2 + b y^2 = z^2 \ </math>
has a point (''x'',''y'',''z'') with coordinates in ''F'' in the split case.<ref name=GS7>Gille & Szamuely (2006) p.7</ref>
==Application==
Quaternion algebras are applied in [[number theory]], particularly to [[quadratic form]]s. They are concrete structures that generate the elements of [[order (group theory)|order]] two in the [[Brauer group]] of ''F''. For some fields, including [[algebraic number
==Classification==
It is a theorem of [[Ferdinand Georg Frobenius|Frobenius]] that there are only two real quaternion algebras: 2&
In a similar way, over any [[local field]] ''F'' there are exactly two quaternion algebras: the 2&
But the quaternion division algebra over a local field is usually ''not'' Hamilton's quaternions over the field. For example, over the [[p-adic number|''p''-adic numbers]] Hamilton's quaternions are a division algebra only when ''p'' is 2. For odd [[prime number|prime]] ''p'', the ''p''-adic Hamilton quaternions are isomorphic to the 2&
:''x''<sup>2</sup> + ''y''<sup>2</sup> = −1
is solvable in the ''p''-adic numbers. Therefore the quaternion
:''xi'' + ''yj'' + ''k''
has norm 0 and hence doesn't have a [[multiplicative inverse]].
One
To every quaternion algebra ''A'', one can associate a quadratic form ''N'' (called the ''[[norm form]]'') on ''A'' such that
Line 47 ⟶ 50:
==Quaternion algebras over the rational numbers==
Quaternion algebras over the [[rational
Let <math>B</math> be a quaternion algebra over <math>\mathbb{Q}</math> and let <math>\nu</math> be a [[Place (mathematics)|place]] of <math>\mathbb{Q}</math>, with completion <math>\mathbb{Q}_\nu</math> (so it is either the ''p''-adic numbers <math>\mathbb{Q}_p</math> for some prime ''p'' or the real numbers <math>\mathbb{R}</math>). Define <math>B_\nu:= \mathbb{Q}_\nu \otimes_{\mathbb{Q}} B</math>, which is a quaternion algebra over <math>\mathbb{Q}_\nu</math>. So there are two choices for
<math>B_\nu</math>:
We say that <math>B</math> is '''split''' (or '''unramified''') at <math>\nu</math> if <math>B_\nu</math> is isomorphic to the 2&
A quaternion algebra over the rationals which splits at <math>\infty</math> is analogous to a [[real quadratic field]] and one which is non-split at <math>\infty</math> is analogous to an [[imaginary quadratic field]].
it is infinite if the quaternion algebra splits at <math>\infty</math>{{Citation needed|date=July 2009}} and it is finite otherwise{{Citation needed|date=July 2009}}, just as the unit group of an order in a quadratic ring is infinite in the real quadratic case and finite otherwise.
The number of places where a quaternion algebra over the rationals ramifies is always even, and this is equivalent to the [[quadratic reciprocity law]] over the rationals.
Moreover, the places where ''B'' ramifies determines ''B'' up to isomorphism as an algebra. (In other words, non-isomorphic quaternion algebras over the rationals do not share the same set of ramified places.) The product of the primes at which ''B'' ramifies is called the '''discriminant''' of ''B''.
==See also==
*[[
*[[
*[[
*[[Hurwitz quaternion order]]
*[[Hurwitz quaternion]]
==
{{reflist}}
* {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}▼
==References==
* {{cite book | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | publisher=[[American Mathematical Society]] | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}▼
▲* {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 | doi=10.1017/CBO9780511607219}}
▲* {{cite book | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | title=Introduction to Quadratic Forms over Fields | volume=67 | series=[[Graduate Studies in Mathematics]] | publisher=[[American Mathematical Society]] | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
==Further reading==
{{wikibooks|Associative Composition Algebra|Quaternions|Quaternion algebras over R and C}}
* {{cite book | last1=Knus | first1=Max-Albert | last2=Merkurjev | first2=Alexander | author2-link=Alexander Merkurjev | last3=Rost | first3=Markus | author3-link=Markus Rost | last4=Tignol | first4=Jean-Pierre | author-link4=Jean-Pierre Tignol | title=The book of involutions | others=With a preface by J. Tits | zbl=0955.16001 | series=Colloquium Publications | publisher=[[American Mathematical Society]] | volume=44 | location=Providence, RI | year=1998 | isbn=0-8218-0904-0 | mr=1632779 }}
* {{cite book| first1=Colin | last1=Maclachlan | first2=Alan W. | last2=Ried | year=2003 | title=The Arithmetic of Hyperbolic 3-Manifolds | isbn=0-387-98386-4 | doi=10.1007/978-1-4757-6720-9 | mr=1937957 | publisher=Springer-Verlag | location=New York}} See chapter 2 (Quaternion Algebras I) and chapter 7 (Quaternion Algebras II).
* {{cite EB1911|wstitle=Algebra}} (''See section on quaternions.'')
* [https://www.encyclopediaofmath.org/index.php/Quaternion_algebra ''Quaternion algebra''] at [[Encyclopedia of Mathematics]].
{{DEFAULTSORT:Quaternion Algebra}}
[[Category:
[[Category:Quaternions|Algebra]]
|