Quaternion Algebra - Quaternion Algebras Over The Rational Numbers

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: it is infinite if the quaternion algebra splits at and it is finite otherwise, 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. 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.