Give an example of fields $k \subset K \subset L$ and $l \subset L$ for which $l/k$ and $L/K$ are algebraic, $k$ is algebraically closed in $K$, and $lK=L$ , but $l$ is not algebraically closed in $L$.
Difficulty in figuring out an example.
$k$ is algebraically closed in $K$, and $lK=L$ , but $l$ is not algebraically closed in $L$.
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abstract-algebra
field-theory
galois-theory
galois-extensions
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0What is the definition of a field being algebraically closed in another field? – 2017-01-26
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0I would say $k$ is algebraically closed in $K$ if every element of $K$ that is algebraic over $k$ is already in $k$. First I thought that means $K$ is purely transcendental over $k$, but that is not true ($K = Frac (k[x,y]/(y^2-x^3)$). I do not see an example either and wonder whether one exists ... – 2017-01-26
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0@user8795: Are you sure there is an example? Do you have partial results? – 2017-01-29