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  "body": "\n## Definition\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] with ring of integers $\\mathcal{O}_K$.\nLet $\\sigma_1, \\ldots, \\sigma_{r_1}$ be the real embeddings of $K$, and let $\\sigma_{r_1 + 1}, \\ldots, \\sigma_{r_1 + r_2}$ be one embedding from each pair of complex conjugate non-real embeddings.\nPut $N_j = 1$ for $j \\le r_1$ and $N_j = 2$ for $j \u003e r_1$, and let $r = r_1 + r_2 - 1$.\nLet $u_1, \\ldots, u_r$ be a system of fundamental units, as in [[Dirichlet's unit theorem|1D99CSKX6N]].\nIf $r \\ge 1$, the *regulator* of $K$ is\n$$R_K = \\abs{\\det\\big(N_j \\log \\abs{\\sigma_j(u_i)}\\big)_{1 \\le i, j \\le r}}.$$\nThis is the absolute value of the [[determinant|00KDZRW4C0]] of an $r \\times r$ matrix.\nThe matrix keeps the first $r$ of the $r + 1$ columns $j = 1, \\ldots, r_1 + r_2$.\nIf $r = 0$, the regulator is $R_K = 1$.\n\n## Remarks\n\nThe value does not depend on the choices.\nEach row of the full $r \\times (r+1)$ matrix sums to $0$, because $\\prod_j \\abs{\\sigma_j(u)}^{N_j} = \\abs{N_{K/\\Q}(u)} = 1$ for a unit $u$; so deleting another column changes only the sign of the determinant.\nAnother system of fundamental units differs by a matrix in $\\mathrm{GL}_r(\\Z)$, which has determinant $\\pm 1$.\nHere $\\log$ is the [[natural logarithm|04DXAFN2ES]].\n\nExample: for $K = \\Q(\\sqrt{2})$, $r = 1$ and $u_1 = 1 + \\sqrt{2}$, so $R_K = \\abs{\\log(1 + \\sqrt{2})} \\approx 0.881$.\n\nNon-example: in $\\Q(\\sqrt{2})$, the unit $3 + 2\\sqrt{2} = (1 + \\sqrt{2})^2$ is not fundamental, since it does not generate the units modulo $\\pm 1$.\nUsing it in the formula gives $2 \\log(1 + \\sqrt{2})$, which is twice the regulator.\n\nFor $\\Q$ and for imaginary quadratic fields, $r = 0$ and $R_K = 1$.\n",
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