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Let $L/K$ be a finite field extension ( possibly separable ) with algebraic closure $\bar{K}$. Let $\sigma : L \to \bar{K}$ be a $K$-embedding. Then $\sigma $ is extendible to a $K$-automorpism $\tilde{\sigma} : \bar{K} \to \bar{K}$? This fact(?) is used in proof of the Theorem 5.1 ( transitivity of the norm and trace ) , p.285, in the Lang's Algebra, 3rd edition and I don't find associated reference about the proof at all until now.

Plantation
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    A variant of this was recently discussed here. I tried to explain the method of applying Zorn's lemma to this end here. We have more lucid writers than yours truly on this site, so there may be something better out there. – Jyrki Lahtonen Apr 22 '24 at 10:15
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    The asker in that latter question later had the assumption of finiteness of the extension. My answer does have the benefit of showing that it is not necessary (that's were Zorn comes in) - to the extent possible. You need to ask a set/model theorist whether the result is true if we don't include the axiom of choice into our repertory. – Jyrki Lahtonen Apr 22 '24 at 10:18
  • @JyrkiLahtonen : Thank you. I think that I understand your overall strategy of the answer in the latter question. ( using Zorn's Lemma ). Fancy proof. I think I need to check it in more detail. And why there is no proof / or as a form of exercise about this kind of statement in existing textbook? And I think that the discussion about the necessity of Zorn's Lemma is an interesting subject. Anyway thank you for your thoughtful comments. – Plantation Apr 22 '24 at 10:35
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    I think Jacobson's Basic Algebra II has it but cannot check right now. BA I does not seem to have it, but I'm fairly sure I learned this kind of stuff from Jacobson's books. – Jyrki Lahtonen Apr 22 '24 at 12:08
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    Basic Algebra II, chapter 8, does have related material, but it is a bit terse at some spots. I certainly didn't come up with the argument. May be I took the glossing over of a few details in BAII as exercises, but I'm fairly sure I picked up the idea from some text :-) – Jyrki Lahtonen Apr 22 '24 at 15:47
  • O.K. Thanks ! ~ – Plantation Apr 23 '24 at 02:22

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