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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 *norm* of $\\alpha$ from $L$ to $K$ is $N_{L/K}(\\alpha) = \\det(m_\\alpha) \\in K$, the [[determinant|03F5KP94K0]] of $m_\\alpha$.\n\n## Remarks\n\nThe norm is multiplicative: $N_{L/K}(\\alpha\\beta) = N_{L/K}(\\alpha) N_{L/K}(\\beta)$, and $N_{L/K}(c) = c^{[L:K]}$ for $c \\in K$.\nIt is transitive in towers: $N_{M/K} = N_{L/K} \\circ N_{M/L}$.\nIf the [[minimal polynomial|0GK1WB9R98]] of $\\alpha$ over $K$ is $x^d + a_{d-1}x^{d-1} + \\cdots + a_0$, then $N_{L/K}(\\alpha) = \\left((-1)^d a_0\\right)^{[L:K(\\alpha)]}$.\nIf $L/K$ is separable and $\\sigma_1, \\ldots, \\sigma_n$ are the embeddings of $L$ over $K$ into an algebraic closure, then $N_{L/K}(\\alpha) = \\sigma_1(\\alpha) \\cdots \\sigma_n(\\alpha)$.\n\nExample: $L = \\Q(i)$, $K = \\Q$ and $\\alpha = a + bi$ with $a, b \\in \\Q$.\nCheck: in the basis $1, i$, the map $m_\\alpha$ sends $1 \\mapsto a + bi$ and $i \\mapsto -b + ai$, so its matrix is $\\begin{pmatrix} a \u0026 -b \\\\ b \u0026 a \\end{pmatrix}$ and $N(\\alpha) = a^2 + b^2$.\n\nNon-example: the complex absolute value $a + bi \\mapsto \\sqrt{a^2 + b^2}$ is not the norm of $\\Q(i)/\\Q$, since it does not take values in $\\Q$; for example $\\abs{1 + i} = \\sqrt{2}$.\n",
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