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  "body": "\n## Summary\n\nThe trace form of a finite extension $L/K$ is the [[symmetric bilinear form|0AXZNE5XNH]] $(x, y) \\mapsto \\mathrm{Tr}_{L/K}(xy)$ on $L$, built from the [[field trace|WPE5Z53QYJ]].\nIt is [[nondegenerate|01NHFZF2SK]] exactly when $L/K$ is separable.\nIts Gram determinant in a basis is the [[discriminant|3BCYVS9E17]] of that basis.\nOver $\\Q$ the trace form is a quadratic form whose signature counts the real embeddings of a number field.\n\n## Explanations\n\n### L1 human\n\nThe trace turns each element of $L$ into an element of $K$.\nApplying it to products $xy$ gives a way to pair two elements of $L$ and get a number in $K$, like a dot product.\nFor $\\Q(i)$ over $\\Q$, the pairing of $a + bi$ with $c + di$ is $2(ac - bd)$.\nIn characteristic $p$ this pairing can vanish completely, and that happens exactly for inseparable extensions.\n",
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