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  "body": "\n## Statement\n\nLet $D$ be a [[division ring|0WZXK05ME7]], and let $K \\subseteq D$ be a division [[subring|0859D58WPD]] of $D$.\nAssume that $x K x^{-1} \\subseteq K$ for every nonzero $x \\in D$.\nThen $K = D$ or $K$ is contained in the [[center|MVT64B3B6H]] $Z(D) = \\set{z \\in D : zd = dz \\text{ for all } d \\in D}$.\n\n## Remarks\n\nA short proof works with two conjugates.\nTake $a \\in K$ and $b \\in D \\setminus K$; then $b \\ne 0$ and $1 + b \\ne 0$, so $k_1 = b a b^{-1}$ and $k_2 = (1+b) a (1+b)^{-1}$ lie in $K$.\nExpanding $(1+b)a = k_2(1+b)$ and using $ba = k_1 b$ gives $(k_1 - k_2) b = k_2 - a$, so $k_1 = k_2$ (otherwise $b \\in K$), hence $a = k_2 = k_1$ and $ab = ba$.\nSo every element of $K$ commutes with $D \\setminus K$; if $K \\ne D$, every element of $K$ is a difference of two elements of $D \\setminus K$, so it also commutes with $K$, and $K \\subseteq Z(D)$.\nExample: in the real quaternions $\\mathbb{H}$, the subfield $\\C = \\R + \\R i$ is neither $\\mathbb{H}$ nor central, and indeed $(1+j)\\, i \\,(1+j)^{-1} = -k \\notin \\C$.\n\n## Explanations\n\n### L1 human\n\nConjugating by $x$ means replacing $y$ by $x y x^{-1}$.\nThe theorem says that a division subring which is stable under all conjugations is either everything or sits inside the center.\nSo a skew field has no \"normal\" sub-skew-fields other than the trivial ones.\n\n### L2 human\n\nThe theorem is a division-ring analogue of the fact that a normal subgroup is a union of conjugacy classes.\nIt is often applied with $K$ the subring generated by a conjugation-invariant set, such as the center together with some commutators.\nIts proof is elementary, in contrast with deeper results such as [[Wedderburn's little theorem|0RRNVX1VKA]], and it has a companion in [[Hua's identity|AWS7XTZ49C]], another elementary tool for division rings.\n",
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  "name": "Cartan–Brauer–Hua theorem",
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