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  "body": "\n## Summary\n\nLet $D$ be a [[division ring|0EA5BTA0CX]] that is finite-dimensional over its center, a field $K$.\nA maximal subfield of $D$ is a subfield that contains $K$ and is not contained in a larger subfield.\nIts degree over $K$ always equals the [[degree|NNXSP5KKAC]] of $D$, and it [[splits|NVD7R0T835]] $D$.\nSo the study of a division algebra reduces in part to the study of the fields it contains.\n\n## Explanations\n\n### L1 human\n\nA division algebra contains many commutative pieces, and some of them are fields.\nA maximal subfield is one of these fields that cannot be made larger inside the algebra.\nIn Hamilton's quaternions over $\\R$, the elements $x + yi$ form a copy of $\\C$, which is a maximal subfield.\nThere are infinitely many such copies, one for each line through $0$ in the space spanned by $i, j, k$.\n",
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