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if $k$ is any field then it is algebraically closed in $k(x)$. What if we take a nontrivial extension of $k$, i.e. $k'/k$, is $k$ still algebraically closed in $k'(x)$, maybe or never ? ($x$ is not algebraic over $k$)

Because there is an argument in a proof if $[E:k]=\infty$ then consider the infinite chain $k\subset E_1\subset E_2\subset\dots$ (distinct subextensions of $E)$ then there is an induced chain $k(x)\subset E_1(x)\subset E_2(x)\subset\dots$. the author argues in the following way:

$E_i$ is algebraically closed in $E_i(x)$ but not in $E_{i+1}(x)$ so new chain also consists of distinct elements.

the purpose of the proof is to show that $[E(x):k(x)]=[E:k]$

Jno
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  • No, $k(x)$ need not be algebraically closed, see here. What is the definition of "$k$ is algebraically closed in $k(x)$". Integrally closed perhaps? – Dietrich Burde Feb 01 '20 at 19:06
  • @DietrichBurde Let $L/k$ be a field extension. The field $k$ is algebraically closed in $L$ if the only elements of $L$ algebraic over $k$ are the elements of $k$ itself – Jno Feb 01 '20 at 19:11
  • I see. Do you have a wikipedia link for this definition? – Dietrich Burde Feb 01 '20 at 19:16
  • what do you mean with that ? if it matters I can call it ''integrally closed'' if you want – Jno Feb 01 '20 at 19:20
  • Having a context would certainly help in getting reasonable answers, this is what I mean with it. I just cannot find the context, so I would be grateful if you could provide it. Integrally closed seems to be different. Of course, I can leave the question to others if you want. – Dietrich Burde Feb 01 '20 at 19:25
  • @DietrichBurde here is the definition https://imgur.com/u0O5r5S and here is where I got stuck https://imgur.com/jRwgqgr – Jno Feb 01 '20 at 19:30

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If $k$ is not algebraically closed in $k'$ (for instance, in the extreme case where $k'/k$ is an algebraic extension) then clearly $k$ is not algebraically closed in $k'(x)$: any $a\in k'$ which is algebraic over $x$ is also an element of $k'(x)$ which is algebraic over $k$.

Captain Lama
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