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  "body": "\n## Definition\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] of degree $n = [K : \\Q]$, and let $\\mathcal{O}_K$ be its ring of integers.\nLet $\\omega_1, \\ldots, \\omega_n$ be an integral basis of $K$, that is, a basis of $\\mathcal{O}_K$ as a free $\\Z$-module.\nLet $\\sigma_1, \\ldots, \\sigma_n$ be the $n$ field embeddings $K \\to \\C$.\nThe *discriminant* of $K$ is the number\n$$d_K = \\Big(\\det\\big(\\sigma_i(\\omega_j)\\big)_{1 \\le i, j \\le n}\\Big)^2,$$\nwhere $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nThe value does not depend on the integral basis or on the order of the embeddings.\nTwo integral bases differ by a matrix in $\\mathrm{GL}_n(\\Z)$, whose determinant is $\\pm 1$.\nReordering the rows changes only the sign of the determinant.\nAn equivalent form is $d_K = \\det\\big(\\operatorname{Tr}_{K/\\Q}(\\omega_i \\omega_j)\\big)$, where the field trace $\\operatorname{Tr}_{K/\\Q}(\\alpha)$ is the [[trace|0KZDNGA561]] of the $\\Q$-linear map $x \\mapsto \\alpha x$ on $K$.\nThe field trace of an algebraic integer is an integer, so $d_K$ is an integer.\nIt is never $0$, and its sign is $(-1)^{r_2}$, where $r_2$ is the number of pairs of complex conjugate embeddings.\n\nExample: for $K = \\Q(i)$ the basis $1, i$ of $\\Z[i]$ gives $\\det \\begin{pmatrix} 1 \u0026 i \\\\ 1 \u0026 -i \\end{pmatrix} = -2i$, so $d_K = -4$.\nMore generally, for a squarefree integer $d \\ne 1$, the discriminant of $\\Q(\\sqrt{d})$ is $d$ if $d \\equiv 1 \\pmod 4$ and $4d$ otherwise.\n\nNon-example: the basis $1, 2i$ gives $\\det \\begin{pmatrix} 1 \u0026 2i \\\\ 1 \u0026 -2i \\end{pmatrix}^2 = (-4i)^2 = -16$.\nThis is not $d_{\\Q(i)}$, because $1, 2i$ is a basis of the subring $\\Z[2i]$ and not of the whole ring of integers $\\Z[i]$.\n",
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