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  "body": "\n## Definition\n\nLet $L/K$ be a [[finite extension|0NRYPM9Y27]] and let $\\alpha \\in L$.\nLet $m_\\alpha \\colon L \\to L$ be the map $x \\mapsto \\alpha x$; it is a [[$K$-linear map|0RJF668P1T]] of the finite-dimensional $K$-vector space $L$.\nThe *trace* of $\\alpha$ from $L$ to $K$ is $\\mathrm{Tr}_{L/K}(\\alpha) = \\tr(m_\\alpha) \\in K$, the [[trace|0RPDTWY2JS]] of $m_\\alpha$.\n\n## Remarks\n\nThe map $\\mathrm{Tr}_{L/K} \\colon L \\to K$ is $K$-linear, and $\\mathrm{Tr}_{L/K}(c) = [L:K]\\, c$ for $c \\in K$.\nIt is transitive in towers: $\\mathrm{Tr}_{M/K} = \\mathrm{Tr}_{L/K} \\circ \\mathrm{Tr}_{M/L}$.\nIf the [[minimal polynomial|0GK1WB9R98]] of $\\alpha$ over $K$ is $x^d + a_{d-1}x^{d-1} + \\cdots + a_0$, then $\\mathrm{Tr}_{L/K}(\\alpha) = -[L:K(\\alpha)]\\, a_{d-1}$.\nIf $L/K$ is separable and $\\sigma_1, \\ldots, \\sigma_n$ are the embeddings of $L$ over $K$ into an algebraic closure, then $\\mathrm{Tr}_{L/K}(\\alpha) = \\sigma_1(\\alpha) + \\cdots + \\sigma_n(\\alpha)$.\nThe map $\\mathrm{Tr}_{L/K}$ is not identically zero if and only if $L/K$ is separable.\n\nExample: $L = \\Q(i)$, $K = \\Q$ and $\\alpha = a + bi$ with $a, b \\in \\Q$.\nCheck: the matrix of $m_\\alpha$ in the basis $1, i$ is $\\begin{pmatrix} a \u0026 -b \\\\ b \u0026 a \\end{pmatrix}$, so $\\mathrm{Tr}(\\alpha) = 2a$.\n\nNon-example: the real part $a + bi \\mapsto a$ is $\\Q$-linear but is not the trace of $\\Q(i)/\\Q$; it is half of it, since $\\mathrm{Tr}(1) = [\\Q(i):\\Q] = 2$.\n",
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