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  "body": "\n## Statement\n\nLet $K \\subseteq L \\subseteq M$ be fields.\nThen\n$$[M : K] = [M : L] \\cdot [L : K].$$\n\n## Remarks\n\nThe proof: let $u_1, \\dots, u_d$ be a basis of $L$ over $K$, and let $w_1, \\dots, w_e$ be a basis of $M$ over $L$.\nThen the products $u_i w_j$ are a basis of $M$ over $K$.\nThe formula also holds for infinite degrees, as a product of cardinals.\nSo $M/K$ is finite if and only if $M/L$ and $L/K$ are finite.\nIf $[M : K]$ is a prime number, then $M/K$ has no [[intermediate field|02NBKVQFRQ]] other than $K$ and $M$.\n\n## Explanations\n\n### L1 human\n\nDegrees multiply along a tower of fields.\nFor example, $[\\mathbb{Q}(\\sqrt{2}, \\sqrt{3}) : \\mathbb{Q}] = 2 \\cdot 2 = 4$.\nThe law is like Lagrange's theorem for groups, and it is the main tool to compute degrees.\n",
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        "url": "https://en.wikipedia.org/wiki/Degree_of_a_field_extension"
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