Quaternion algebra: Difference between revisions

In [[mathematics]], a '''quaternion algebra''' over a [[field (mathematics)|field]] ''F'' is a [[central simple algebra]] ''A'' over ''F''<ref>See PeircePierce. Associative algebras. Springer. Lemma at page 14.</ref><ref>See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2.</ref> that has [[dimension (vector space)|dimension]] 4 over ''F''. Every quaternion algebra becomes thea [[matrix algebra]] by ''[[Scalar extension|extending scalars]]'' (=equivalently, [[tensor product of algebras|tensoring]] with a [[field extension]]), i.e. for a suitable [[field extension]] ''K'' of ''F'', <math>A \otimes_F K</math> is [[isomorphic]] to the 2×22&thinsp;×&thinsp;2 [[matrix algebra]] over ''K''.

The notion of a quaternion algebra can be seen as a generalization of theHamilton's [[Hamilton quaternionsquaternion]]s to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over <math>F = \mathbb{R}</math> (the [[real number field]]), and indeed the only one over ℝ<math>\mathbb{R}</math> apart from the 2×22&thinsp;×&thinsp;2 [[real number|real]] matrix algebra, up to isomorphism. When <math>F = \mathbb{C}</math>, then the [[biquaternion]]s form the quaternion algebra over ''F''.

''Quaternion algebra'' here means something more general than the [[algebra over a field|algebra]] of [[Hamilton's quaternions[[quaternion]]s. When the coefficient [[field (mathematics)|field]] ''F'' does not have [[characteristic (algebra)|characteristic]] 2, every quaternion algebra over ''F'' can be described as a 4-dimensional ''F''-[[vector space]] with [[basis (linear algebra)|basis]] <math>\{ 1, i, j, k\}</math>, with the following multiplication rules:

:<math>ji=-k</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

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 fieldsfield]]s, every element of order 2 in its Brauer group is represented by a quaternion algebra. A theorem of [[Alexander Merkurjev]] implies that each element of order 2 in the Brauer group of any field is represented by a [[tensor product]] of quaternion algebras.<ref name=Lam139>Lam (2005) p.139</ref> In particular, over [[p-adic field|''p''-adic field]]s the construction of quaternion algebras can be viewed as the quadratic [[Hilbert symbol]] of [[local class field theory]].

It is a theorem of [[Ferdinand Georg Frobenius|Frobenius]] that there are only two real quaternion algebras: 2&timesthinsp;×&thinsp;2 matrices over the reals and Hamilton's real quaternions.

In a similar way, over any [[local field]] ''F'' there are exactly two quaternion algebras: the 2&timesthinsp;×&thinsp;2 matrices over ''F'' and a division algebra.

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&timesthinsp;×&thinsp;2 matrices over the ''p''-adics. To see the ''p''-adic Hamilton quaternions are not a division algebra for odd prime ''p'', observe that the [[modular arithmetic|congruence]] ''x''² + ''y''² = −1 mod ''p'' is solvable and therefore by [[Hensel's lemma]] &mdash;— here is where ''p'' being odd is needed &mdash;— the equation

:''x''² + ''y''² = −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 would likeway to classify the [[Algebra homomorphism|''F''-algebra isomorphism]] [[equivalenceisomorphism class|classes]]es of all quaternion algebras for a given field, ''F''. One way to do this is to use the one-to-one correspondence between isomorphism classes of quaternion algebras over ''F'' and isomorphism classes of their ''norm forms''.

Quaternion algebras over the [[rational numbersnumber]]s have an arithmetic theory similar to, but more complicated than, that of [[quadratic field|quadratic extensions of <math>\mathbb{Q}</math>]].

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>: the 2 by &thinsp;×&thinsp;2 matrices over <math>\mathbb{Q}_\nu</math> or a [[division algebra]].

We say that <math>B</math> is '''split''' (or '''unramified''') at <math>\nu</math> if <math>B_\nu</math> is isomorphic to the 2&timesthinsp;×&thinsp;2 matrices over <math>\mathbb{Q}_\nu</math>. We say that ''B'' is '''non-split''' (or '''ramified''') at <math>\nu</math> if <math>B_\nu</math> is the quaternion division algebra over <math>\mathbb{Q}_\nu</math>. For example, the rational Hamilton quaternions is non-split at 2 and at <math>\infty</math> and split at all odd primes. The rational 2 by &thinsp;×&thinsp;2 matrices are split at all places.

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]]. The analogy comes from a quadratic field having real embeddings when the [[minimal polynomial (field theory)|minimal polynomial]] for a generator splits over the reals and having non-real embeddings otherwise. One illustration of the strength of this analogy concerns [[unit groupsgroup]]s in an order of a rational quaternion algebra:

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.

*[[compositionComposition algebra]]

*[[cyclicCyclic algebra]]

*[[octonionOctonion algebra]]

==ReferencesNotes==

* {{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 }}

[[Category:AlgebrasComposition algebras]]