Quaternion algebra

Remarks

When the characteristic of $F$ is not $2$, every quaternion algebra over $F$ is isomorphic to $\left(\frac{a, b}{F}\right)$ for some nonzero $a, b$. In this algebra $k^2 = -ab$.

Example: $\left(\frac{-1, -1}{\R}\right)$ is Hamilton's algebra $\mathbb{H}$ of quaternions. The element $q = x + yi + zj + wk$ has $q \bar{q} = x^2 + y^2 + z^2 + w^2$ for $\bar{q} = x - yi - zj - wk$, so every $q \ne 0$ is invertible, and $\mathbb{H}$ is a division algebra.

Example: $\left(\frac{1, 1}{F}\right) \cong M_2(F)$, through $i \mapsto \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ and $j \mapsto \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.

Non-example: the algebra $F[x, y]/(x^2 - a, y^2 - b, xy - yx)$ also has dimension $4$, but it is commutative, so its center is not $F$ and it is not a quaternion algebra.

Every quaternion algebra is either a division algebra or isomorphic to $M_2(F)$. The algebra $\left(\frac{a, b}{F}\right)$ is isomorphic to $M_2(F)$ if and only if the equation $a x^2 + b y^2 = 1$ has a solution with $x, y \in F$.