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  "body": "\n## Summary\n\nLet $L/K$ be a [[finite extension|0C1Y04SZ1W]] and let $\\alpha \\in L$.\nMultiplication by $\\alpha$ is a $K$-linear map of $L$, and the field trace $\\mathrm{Tr}_{L/K}(\\alpha)$ is its [[trace|0KZDNGA561]].\nIt is a $K$-linear map $L \\to K$.\nFor a separable extension it is the sum of the images of $\\alpha$ under all embeddings of $L$ into an algebraic closure.\nThe trace gives the [[trace form|YWX2R53ED6]] and the [[discriminant|3BCYVS9E17]] of an extension.\nThe trace map is not identically zero exactly when the extension is separable.\n\n## Explanations\n\n### L1 human\n\nMultiplying by a fixed element $\\alpha$ is a linear map of the field $L$, seen as a vector space over $K$.\nThe trace of $\\alpha$ is the trace of this map, the sum of the diagonal entries of its matrix.\nFor Gaussian numbers $a + bi$ over $\\Q$, the trace is $(a + bi) + (a - bi) = 2a$.\nSo the trace adds up all the conjugates of $\\alpha$, while the [[norm|KRNFEZWPY0]] multiplies them.\n",
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