Quaternion algebra

From Wikipedia, the free encyclopedia

In mathematics, a quaternion algebra over a field F is a particular kind of central simple algebra A over F, namely such an algebra that has dimension 4, and therefore becomes the 2×2 matrix algebra over some field extension of F, by extending scalars (i.e. tensoring with a field extension). The classical quaternions are the case of F the real number field, and A is uniquely defined up to isomorphism by the condition that it is such a quaternion algebra that is not the 2×2 real matrix algebra.

Quaternion algebra therefore means something other than the algebra of quaternions. In fact when the base field F does not have characteristic 2, any quaternion algebra over F is a slightly twisted form of the familiar quaternions, with a basis 1, i, j, and k such that

i² = a

j² = b

ij = k, ji = −k

where a and b are nonzero elements of F and need not be −1. 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 description as a 4-dimensional central simple algebra applies uniformly in all characteristics.

Quaternion algebras are applied in number theory, particularly to quadratic forms. They are concrete structures that represent the elements of order two in the Brauer group. In particular, for p-adic fields, their theory includes that of the Hilbert symbol of local class field theory.

[edit] Quaternion algebras over the rational numbers

Quaternion algebras over the rational numbers have an arithmetic theory similar to, but more complicated than, that of algebraic number fields, more precisely of quadratic field extensions of $\mathbb{Q}$ .

Let B be a quaternion algebra over the field $\mathbb{Q}$ of rational numbers. Let $ν$ be a place of $\mathbb{Q}$ , and let $\mathbb{Q}_\nu$ denote the completion of $\mathbb{Q}$ at $ν$ (so either a p-adic field $\mathbb{Q}_p$ or the real numbers $\mathbb{R}$ ). We define $B_\nu:= B\otimes_{\mathbb{Q}}\mathbb{Q}_\nu$ .

We say that B is split at $ν$ if $B ν$ is isomorphic to the algebra of 2×2 matrices over $\mathbb{Q}_\nu$ . We say that B is ramified at $ν$ if $B ν$ is the quaternion division algebra over $\mathbb{Q}_\nu$ . The set of ramified places is always finite, and it determines B up to isomorphism. The product of the primes at which B is ramified is called the discriminant of B.

Retrieved from "http://en.wikipedia.org../../../q/u/a/Quaternion_algebra.html"

Categories: Ring theory | Quaternions

Quaternion algebra

From Wikipedia, the free encyclopedia

[edit] Quaternion algebras over the rational numbers

Views

Navigation

interaction

Search