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  "body": "\n## Statement\n\nLet $L/K$ be a [[finite extension|0NRYPM9Y27]], and let $\\Aut(L/K)$ be the group of [[automorphisms|0K1SCPBNKJ]] of $L$ that fix $K$ pointwise.\nThe following are equivalent:\n\n1. $L/K$ is [[normal|0CBRFQ7DVZ]] and [[separable|0N019F6RE9]];\n2. $L$ is the [[splitting field|02NZZ1J0FH]] over $K$ of a [[separable polynomial|03J8BS7FMV]] in $K[x]$;\n3. $\\abs{\\Aut(L/K)} = [L:K]$, where $[L:K]$ is the [[degree|0GS74FGFEK]];\n4. the [[fixed field|0KMJ6919WS]] of $\\Aut(L/K)$ is $K$;\n5. $K$ is the fixed field of some finite group of automorphisms of $L$.\n\nAn extension with these properties is a Galois extension.\n\n## Remarks\n\nThe implication from 1 to 2 takes the product of the distinct minimal polynomials of a basis.\nFrom 2 to 3, one counts the extensions of the identity of $K$ to automorphisms of $L$ step by step, using that the roots are distinct.\nIn general $\\abs{\\Aut(L/K)} \\le [L:K]$, and [[Artin's theorem on fixed fields|X7R15C00ET]] gives $[L:L^{G}] = \\abs{G}$ for $G = \\Aut(L/K)$, which links 3, 4 and 5.\nFrom 5 to 1, every element of $L$ is a root of the separable split polynomial $\\prod (x - \\beta)$ over its orbit.\n\nExample: $\\Q(\\sqrt[3]{2})/\\Q$ is not Galois.\nIts only automorphism is the identity, since the other roots of $x^3 - 2$ are not real; so condition 3 fails ($1 \\ne 3$) and condition 4 fails (the fixed field is $\\Q(\\sqrt[3]{2})$).\n\nFiniteness is needed for condition 3 to make sense; for infinite algebraic extensions, conditions 1 and 4 remain equivalent.\n\n## Explanations\n\n### L1 human\n\nA Galois extension is a field extension with as much symmetry as possible.\nThis theorem lists several tests for it.\nOne test: each irreducible polynomial with one root in $L$ has all its roots in $L$, with no repeated roots.\nAnother test: count the symmetries and compare with the degree.\nA third test: only the elements of $K$ are fixed by all symmetries.\n\n### L2 human\n\nIn practice one shows that an extension is Galois by exhibiting it as a splitting field of a separable polynomial, condition 2.\nOne uses condition 3 or 4 to compute with the Galois group, for example to show that $\\abs{\\Gal(L/K)} = [L:K]$.\nCondition 5 is how Galois extensions arise from group actions.\nThese equivalences are the input for the [[fundamental theorem of Galois theory|0T1Q8D0FW8]].\n",
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