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  "body": "\n## Summary\n\nLet $E$ and $F$ be extensions of a field $K$.\nTheir tensor product $E \\otimes_K F$ is the [[tensor product of vector spaces|0BF2DE2SYK]] over $K$, with the product $(a \\otimes b)(c \\otimes d) = ac \\otimes bd$.\nIt is usually not a field.\nEach way to put $E$ and $F$ inside one field gives a quotient of $E \\otimes_K F$ by a prime ideal.\nThe field of fractions of this quotient is the [[compositum|T1S29NN8PE]], so the tensor product records all these ways at once.\nFor a finite separable extension $E = K(\\alpha)$ it is a product of fields, one for each irreducible factor over $F$ of the minimal polynomial of $\\alpha$.\n\n## Explanations\n\n### L1 human\n\nGiven two extensions of $K$, there may be several ways to fit them together in a larger field.\nThe tensor product keeps track of all of them at once.\nFor example, $\\C \\otimes_\\R \\C$ is isomorphic to $\\C \\times \\C$.\nThe two factors correspond to the two ways of identifying the second copy of $\\C$ with the first: the identity and complex conjugation.\n",
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