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  "body": "\n## Summary\n\nLet $E$ and $F$ be extensions of a field $K$ inside a common field.\nThey are linearly disjoint over $K$ when elements of $E$ that are [[linearly independent|0CRMGPAH83]] over $K$ stay independent over $F$.\nEquivalently, the natural map from the [[tensor product|ANH16GQJZQ]] $E \\otimes_K F$ to the common field is injective.\nFor finite extensions this means that the degree of the [[compositum|T1S29NN8PE]] is as large as possible: $[EF:F] = [E:K]$.\nThe notion is the field version of independence, and it is used to compute degrees and Galois groups of composita.\n\n## Explanations\n\n### L1 human\n\nTwo extensions are linearly disjoint when they do not overlap in a hidden way.\nA basis of one stays a basis after you extend the scalars by the other.\nFor example, $\\Q(\\sqrt{2})$ and $\\Q(\\sqrt{3})$ are linearly disjoint over $\\Q$: together they generate a field of degree $2 \\cdot 2 = 4$.\nHaving only $\\Q$ in common is not enough in general.\nThe fields $\\Q(\\sqrt[3]{2})$ and $\\Q(\\omega\\sqrt[3]{2})$, with $\\omega = e^{2\\pi i/3}$, meet only in $\\Q$, but together they generate a field of degree $6$, not $9$.\n",
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