It is an attempt to construct a division algebra of degree 4 over a given field \(F\) of odd characteristic.

The best reference on this topic is Voight [1].

These notes are really good.

There is also [2] that claims a resolution of what quaternion algebras are division rings over towers of quadratic number fields.

Hamiltonian quaternions

It is given by \[\begin{equation} \mathbb{R} + \mathbb{R} i + \mathbb{R} j + \mathbb{R} k \end{equation}\] where \(i^2 = j^{2} = k^{2} = ijk = -1\). This gives a division algebra over \(\mathbb{R}\). This

PARI-GP

The software gp has some good support for quaternion algebras. See this page.

Here is a link to the official instructions.

References