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  "body": "\n## Definition\n\nLet $F$ be a [[field|04C5ZBG1WT]].\nA *quaternion algebra* over $F$ is a [[central simple algebra|5HRKJBAY2V]] over $F$ of dimension $4$.\n\nLet the [[characteristic|0F6ES3BRYP]] of $F$ be different from $2$, and let $a, b \\in F$ be nonzero.\nThe quaternion algebra $\\left(\\frac{a, b}{F}\\right)$ is the [[associative algebra|3CTW1P4HBP]] over $F$ with basis $1, i, j, k$ and multiplication determined by\n$$i^2 = a, \\qquad j^2 = b, \\qquad ij = -ji = k.$$\n\n## Remarks\n\nWhen the characteristic of $F$ is not $2$, every quaternion algebra over $F$ is isomorphic to $\\left(\\frac{a, b}{F}\\right)$ for some nonzero $a, b$.\nIn this algebra $k^2 = -ab$.\n\nExample: $\\left(\\frac{-1, -1}{\\R}\\right)$ is Hamilton's algebra $\\mathbb{H}$ of quaternions.\nThe 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.\n\nExample: $\\left(\\frac{1, 1}{F}\\right) \\cong M_2(F)$, through $i \\mapsto \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 -1 \\end{pmatrix}$ and $j \\mapsto \\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$.\n\nNon-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.\n\nEvery quaternion algebra is either a division algebra or isomorphic to $M_2(F)$.\nThe 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 $x, y \\in F$.\n",
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