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  "body": "\n## Example\n\nLet $\\eta = 2\\cos(2\\pi/7)$ and let $K = \\Q(\\eta)$, the real cubic subfield of the seventh cyclotomic field, an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\\Z[\\eta]$.\nThe number $\\eta$ is a root of $t^3 + t^2 - 2t - 1$.\nLet $D$ be the [[quaternion algebra|ZWXW0T3EAC]] $(\\eta, \\eta)_K$, with $K$-basis $1, i, j, ij$ and relations $i^2 = j^2 = \\eta$ and $ij = -ji$.\nPut $\\tau = 1 + \\eta + \\eta^2$ and $j' = \\tfrac{1}{2}(1 + \\eta i + \\tau j)$.\nThe *Hurwitz quaternion order* is the $\\Z[\\eta]$-[[order|1ZBX4TKVZ9]]\n$$\\mathcal{Q}_{\\mathrm{Hur}} = \\Z[\\eta][i, j, j']$$\ngenerated by $i$, $j$ and $j'$.\nIt is a maximal order of $D$.\n\nA check that $j'$ is integral: an element $a + bi + cj + d\\,ij$ of $D$ has reduced trace $2a$ and reduced norm $a^2 - \\eta b^2 - \\eta c^2 + \\eta^2 d^2$.\nSo $j'$ has reduced trace $1$ and reduced norm $\\tfrac{1}{4}(1 - \\eta^3 - \\eta\\tau^2)$.\nUsing $\\eta^3 = -\\eta^2 + 2\\eta + 1$ one finds $\\eta \\tau^2 = \\eta^2 + 10\\eta + 4$, so the reduced norm is $\\tfrac{1}{4}(-4 - 12\\eta) = -1 - 3\\eta \\in \\Z[\\eta]$.\nHence $j'$ satisfies the monic equation $x^2 - x - 1 - 3\\eta = 0$ over $\\Z[\\eta]$.\n\n## Remarks\n\nThe group of elements of reduced norm $1$ in $\\mathcal{Q}_{\\mathrm{Hur}}$, modulo $\\set{\\pm 1}$, is isomorphic to the $(2,3,7)$ triangle group.\nQuotients of the hyperbolic plane by its principal congruence subgroups give Hurwitz surfaces, the compact Riemann surfaces with the largest possible automorphism group for their genus.\n\n## Explanations\n\n### L1 human\n\nThis is a specific ring of quaternion-like numbers with coefficients from the field generated by $2\\cos(2\\pi/7)$.\nIt plays the role that the integers play inside the rationals, and it cannot be enlarged to a bigger such ring.\nIts unit group acts on the hyperbolic plane like a very symmetric tiling by triangles with angles $\\pi/2$, $\\pi/3$ and $\\pi/7$.\nThis is why it is used to build the most symmetric Riemann surfaces, such as the Klein quartic.\n",
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