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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 norm $N_{L/K}(\\alpha)$ is its [[determinant|0X0HQYSZ0K]].\nIt is a multiplicative map $L \\to K$, and it is nonzero exactly on the nonzero elements.\nFor a separable extension it is the product of the images of $\\alpha$ under all embeddings of $L$ into an algebraic closure.\nNorms measure the size of algebraic numbers, detect units in rings of integers, and appear in [[Hilbert's Theorem 90|YFF8B5A0QG]].\n\n## Explanations\n\n### L1 human\n\nMultiplying by a fixed element $\\alpha$ stretches the field $L$, seen as a vector space over $K$.\nThe norm is the determinant of this stretching.\nFor the Gaussian numbers $a + bi$ over $\\Q$, it is $a^2 + b^2$, the product of $a + bi$ and its conjugate $a - bi$.\nSo $N(3 + 4i) = 25$.\nThe rule $N(\\alpha\\beta) = N(\\alpha) N(\\beta)$ explains why a product of sums of two squares is again a sum of two squares.\n",
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