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Let $K$ be a field. I would like to characterize $K$ for which every algebraic extension is normal.
It is easy to see that if $K$ is finite(Prove that every extension of a finite field is normal), real closed or algebraicly closed then all of its algebraic extensions are normal. I suspect that these are the only such fields, but I haven't been able to prove it. Could someone help me prove my belief or find a counterexample?

J. W. Tanner
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    $\mathbb{C}[[t]],$ the field of Laurent series in $t$ is a counterexample. It's not finite, algebraically closed or real closed ($-1$ has no square root) and all of its extensions are Abelian, as can be seen from a valuation theoretic argument. – Zsombor Kiss Apr 04 '25 at 19:01
  • @QiaochuYuan: how do you get there? The condition is that every closed subgroup of the absolute Galois group is normal, but that doesn’t obviously imply that the absolute Galois group is abelian – there are well-known counter-examples for finite groups. Can we use the lack of elements of finite order to improve on this? – Aphelli Apr 05 '25 at 09:44
  • @Aphelli: you're right, I was careless. – Qiaochu Yuan Apr 05 '25 at 17:46

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