Quaternion algebra

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 Hamilton 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.

tructure

"Quaternion algebra" therefore means something more general than the "algebra of Hamilton's quaternions". When the coefficient field "F" does not have characteristic 2, any quaternion algebra over "F" is a slightly twisted form of the familiar quaternions with coefficients in "F". It has a basis 1, "i", "j", and "k" such that

:"i"² = "a" :"j"² = "b" : "ij" = "k", "ji" = −"k"

where "a" and "b" are any nonzero elements of "F", and a short calculation shows "k"² = −"ab". (The Hamilton quaternions are the case when "a" and "b" both equal −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 definition of a quaternion algebra over "F" as a 4-dimensional central simple algebra over "F" applies uniformly in all characteristics.

Application

Quaternion algebras are applied in number theory, particularly to quadratic forms. They are concrete structures that generate the elements of index two in the Brauer group of "F". (For some fields, including algebraic number fields, every element of index 2 in its Brauer group is represented by a quaternion algebra. A theorem of Merkurjev says the elements of index 2 in the Brauer group of any field are represented by a tensor product of quaternion algebras.) 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.

"Warning": 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 x² + y² = −1 mod "p" is solvable and therefore by Hensel's lemma here is where "p" being odd is needed the equation

:x² + y² = −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

:$N(xy)\; =\; N(x)N(y)$

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 $mathbb\{Q\}$.

Let "B" be a quaternion algebra over $mathbb\{Q\}$ and let $u$ be a place of $mathbb\{Q\}$, with completion $mathbb\{Q\}\_\; u$ (so it is either the "p"-adic numbers$mathbb\{Q\}\_p$ for some prime "p" or the real numbers $mathbb\{R\}$). Define $B\_\; u:=\; mathbb\{Q\}\_\; u\; otimes\_\{mathbb\{Q\; B$, which is a quaternion algebra over $mathbb\{Q\}\_\; u$. So there are two choices for $B\_\; u$: the 2 by 2 matrices over $mathbb\{Q\}\_\; u$ or a division algebra.

We say that "B" is split (or unramified) at $u$ if $B\_\; u$ is isomorphic to the 2×2 matrices over $mathbb\{Q\}\_\; u$. We say that "B" is non-split (or ramified) at $u$ if $B\_\; u$ is the quaternion division algebra over $mathbb\{Q\}\_\; u$. For example, the rational Hamilton quaternions is non-split at 2 and at $infty$ 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 $infty$ is analogous to a real quadratic field and one which is non-split at $infty$ 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: it is infinite if the quaternion algebra splits at $infty$ and it is finite otherwise, just as an order in a quadratic ring is infinite in the real quadratic case and finite otherwise.

The set of places where a quaternion algebra over the rationals splits is always even, and this is equivalent to the quadratic reciprocity law over the rationals.Moreover, the places where "B" splits 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 split places.) The product of the primes at which "B" splits is called the discriminant of "B".

ee also

*composition algebra
*octonion algebra
*Hurwitz quaternion order