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I have seen this claim multiple times, but every time I read a proof, the first step is to find the minimal polynomial for an element of the field and use its irreducibility to show separability. However, is this not assuming that the field extension is algebraic? Are there non-algebraic extensions of say $\Bbb Q$ that are not separable?

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    $\begingroup$ Separability has a meaning only for algebraic extensions. $\endgroup$ Feb 21 at 0:10
  • $\begingroup$ Actually, an arbitrary field extension is called separable if every finitely generated subextension has a separating transcendence basis, which is a transcendence basis over which the extension is separable in the better known sense. This condition always holds if the base field has characteristic zero, but not if the characteristic is positive. $\endgroup$ Feb 21 at 0:48

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