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  "body": "\n## Definition\n\nLet $K$ be a field and let $D$ be a [[division ring|0EA5BTA0CX]] that is a [[central simple algebra|5HRKJBAY2V]] over $K$.\nA [[subfield|0GFGHRT2FJ]] of $D$ is a subring $L$ of $D$ that is a field.\nA *maximal subfield* of $D$ is a subfield $L$ with $K \\subseteq L$ that is not properly contained in any other subfield of $D$.\n\n## Remarks\n\nEvery maximal subfield $L$ satisfies $[L : K] = \\deg D$, the [[degree|NNXSP5KKAC]] of $D$, and $L$ [[splits|NVD7R0T835]] $D$.\nIndeed, maximality gives $C_D(L) = L$, and the double centralizer theorem gives $[L : K]^2 = \\dim_K D$.\nConversely, a subfield $L \\supseteq K$ with $[L : K] = \\deg D$ is maximal.\nFor a central simple algebra $A$ that is not a division algebra, a subfield $L \\supseteq K$ of $A$ with $[L : K] = \\deg A$ is also called a maximal subfield.\n\nExample: in $\\mathbb{H}$ over $\\R$, the subalgebra $L = \\R + \\R i$ is a field isomorphic to $\\C$, and $[L : \\R] = 2 = \\deg \\mathbb{H}$, so $L$ is a maximal subfield.\n\nNon-example: the subfield $\\R$ of $\\mathbb{H}$ contains $\\R$ but is properly contained in $\\R + \\R i$, so it is not maximal.\n",
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