Quaternion Algebra - Classification

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 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

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.