# Quaternion algebra

In mathematics, a **quaternion algebra** over a field *F* is a central simple algebra *A* over *F*^{[1]}^{[2]} that has dimension 4 over *F*. Every quaternion algebra becomes the matrix algebra by *extending scalars* (=tensoring with a field extension), i.e. for a suitable field extension *K* of *F*, is isomorphic to the 2×2 matrix algebra over *K*.

The notion of a quaternion algebra can be seen as a generalization of the Hamilton quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over (the real number field), and indeed the only one over apart from the 2×2 real matrix algebra, up to isomorphism.

## Structure

*Quaternion algebra* here means something more general than the algebra of Hamilton quaternions. When the coefficient field *F* does not have characteristic 2, every quaternion algebra over *F* can be described as a 4-dimensional *F*-vector space with basis , with the following multiplication rules:

where *a* and *b* are any given nonzero elements of *F*. From these rules we get:

(The Hamilton quaternions are the case where and *a* = *b* = −1.) The algebra defined in this way is denoted (*a*,*b*)_{F} or simply (*a*,*b*).^{[3]} 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*)_{F} is either a division algebra or isomorphic to the matrix algebra of 2×2 matrices over *F*: the latter case is termed *split*.^{[4]} The *norm form*

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 *C*(*a*,*b*) defined by

has a point (*x*,*y*,*z*) with coordinates in *F* in the split case.^{[5]}

## Application

Quaternion algebras are applied in number theory, particularly to quadratic forms. 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.^{[6]} In particular, over p-adic fields the construction of quaternion algebras can be viewed as the quadratic Hilbert symbol of local class field theory.

## Classification

It is a theorem of Frobenius that there are only two real quaternion algebras: 2×2 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 2×2 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 numbers Hamilton's quaternions are a division algebra only when *p* is 2. For odd prime *p*, the *p*-adic Hamilton quaternions are isomorphic to the 2×2 matrices over the *p*-adics. To see the *p*-adic Hamilton quaternions are not a division algebra for odd prime *p*, observe that the congruence *x*^{2} + *y*^{2} = −1 mod *p* is solvable and therefore by Hensel's lemma — here is where *p* being odd is needed — 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 like to classify the *F*-algebra isomorphism classes 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*.

To every quaternion algebra *A*, one can associate a quadratic form *N* (called the *norm form*) on *A* such that

for all *x* and *y* in *A*. It turns out that the possible norm forms for quaternion *F*-algebras are exactly the Pfister 2-forms.

## Quaternion algebras over the rational numbers

Quaternion algebras over the rational numbers have an arithmetic theory similar to, but more complicated than, that of quadratic extensions of .

Let be a quaternion algebra over and let be a place of , with completion (so it is either the *p*-adic numbers for some prime *p* or the real numbers ). Define , which is a quaternion algebra over . So there are two choices for
: the 2 by 2 matrices over or a division algebra.

We say that is **split** (or **unramified**) at if is isomorphic to the 2×2 matrices over . We say that *B* is **non-split** (or **ramified**) at if is the quaternion division algebra over . For example, the rational Hamilton quaternions is non-split at 2 and at and split at all odd primes. The rational 2 by 2 matrices are split at all places.

A quaternion algebra over the rationals which splits at is analogous to a real quadratic field and one which is non-split at is analogous to an imaginary quadratic field. The analogy comes from a quadratic field having real embeddings when the 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 groups in an order of a rational quaternion algebra:
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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

## References

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## Further reading

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