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  "body": "\n## Summary\n\nThe compositum of two subfields $E$ and $F$ of a field $\\Omega$ is the smallest subfield of $\\Omega$ that contains both, written $EF$.\nIt is the field [[generated|05NAS7AGMZ]] by $E \\cup F$, and when $E = K(\\alpha_1, \\ldots, \\alpha_n)$ it equals $F(\\alpha_1, \\ldots, \\alpha_n)$.\nComposita appear throughout Galois theory: the compositum of two Galois extensions is Galois, and its degree is controlled by [[linear disjointness|N6YZFG66YP]].\nWithout a common field $\\Omega$ the compositum is not defined, and the [[tensor product of fields|ANH16GQJZQ]] describes all possible ways to form one.\n\n## Explanations\n\n### L1 human\n\nPut two fields side by side inside one big field.\nTheir union is usually not a field, because sums like $\\sqrt{2} + \\sqrt{3}$ are missing.\nThe compositum is what you get by allowing all sums, products and quotients of elements from both.\nFor $\\Q(\\sqrt{2})$ and $\\Q(\\sqrt{3})$ inside $\\C$, the compositum is $\\Q(\\sqrt{2}, \\sqrt{3})$, a field of degree $4$ over $\\Q$.\n",
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