Quaternion algebra: Difference between revisions

Quaternion algebras are applied in [[number theory]], particularly to [[quadratic form]]s. They are concrete structures that generate the elements of order two in the [[Brauer group]] of ''F''. For some fields, including algebraic number fields, 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]]s the construction of quaternion algebras can be viewed as the quadratic [[Hilbert symbol]] of [[local class field theory]].