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  "body": "\n## Summary\n\nThe discriminant of a basis $b_1, \\ldots, b_n$ of a finite extension $L/K$ is the determinant of the matrix $\\left(\\mathrm{Tr}_{L/K}(b_i b_j)\\right)$, the Gram matrix of the [[trace form|YWX2R53ED6]].\nChanging the basis multiplies it by the square of a nonzero element of $K$, so its class modulo squares depends only on the extension.\nIt is nonzero exactly when $L/K$ is separable.\nFor a power basis $1, \\alpha, \\ldots, \\alpha^{n-1}$ it equals the [[discriminant of the minimal polynomial|0VVK64Q4JK]] of $\\alpha$.\nFor an integral basis of a number field it gives the discriminant of the number field.\n\n## Explanations\n\n### L1 human\n\nPick a basis of the extension and pair basis elements with the trace form.\nThe determinant of the resulting square table is the discriminant.\nIt is nonzero exactly when the trace pairing is nondegenerate.\nFor $\\Q(i)$ with basis $1, i$, the table is $\\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 -2 \\end{pmatrix}$, so the discriminant is $-4$.\nFor the basis $1, \\alpha$ of $\\Q(\\alpha)$ with $\\alpha^2 = d$, it is $4d$, the discriminant $b^2 - 4ac$ of $x^2 - d$.\n",
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