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  "body": "\n## Definition\n\nLet $L/K$ be an [[algebraic extension|0PTBM7K0R2]].\nA *normal closure* of $L/K$ is a field extension $M/L$ such that:\n\n1. $M/K$ is a [[normal extension|0CBRFQ7DVZ]];\n2. no proper subfield of $M$ that contains $L$ is normal over $K$.\n\n## Remarks\n\nA normal closure exists and is unique up to an isomorphism of fields that is the identity on $L$.\nLet $L = K(\\alpha_1, \\ldots, \\alpha_n)$ be finite over $K$, and let $f_i$ be the [[minimal polynomial|0GK1WB9R98]] of $\\alpha_i$ over $K$.\nThen a normal closure is the [[splitting field|0KXQKX9P63]] of $f_1 \\cdots f_n$ over $L$, and it is again finite over $K$.\nLet $N$ be a normal extension of $K$ that contains $L$.\nInside $N$, the normal closure is the compositum of the fields $\\tau(L)$, for all embeddings $\\tau \\colon L \\to N$ over $K$.\n\nExample: for $L = \\Q(\\sqrt[3]{2})$ over $\\Q$, let $M = \\Q(\\sqrt[3]{2}, \\omega)$ with $\\omega = e^{2\\pi i/3}$.\nCheck: $M$ is the splitting field of $x^3 - 2$ over $\\Q$, so it is normal over $\\Q$.\nA normal subfield containing $\\sqrt[3]{2}$ contains all roots $\\sqrt[3]{2}, \\omega\\sqrt[3]{2}, \\omega^2\\sqrt[3]{2}$ of its minimal polynomial, hence contains $M$.\n\nNon-example: an algebraic closure $\\overline{\\Q}$ of $\\Q$ contains $L$ and is normal over $\\Q$, but it fails clause 2, since $M$ is a proper normal subfield containing $L$.\n",
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