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# Be the infinite extension $L / F / K$ if $L / F$ and $F / K$ are separable, then $L / K$ separable?

2017-12-03 03:04:36

For the case where $L / K$ is a finite extension and F an intermediate field I can show that if $L / F$ and $F / K$ are separable then $L / K$ is separable.

Now assuming $L / K$ an infinite extension and $F$ an intermediate field, if $L / F$ and $F / K$ are separable, $L / K$ will be separable? Is it possible to display some examples that are not worth?

$x\in L\Rightarrow$ the minimal polynomial $x^n+a_{n-1}x^{n-1}+...+a_0$ of $x$ over $F$ is seperable $\Rightarrow K[a_0,...,a_{n-1},x]/K[a_0,...,a_{n-1}]$ and $K[a_0,...,a_{n-1}]/K$ seperable.

First arrow: $L/F$ is seperable

Second arrow: $K[x_1,...,x_n]/K$ seperable $\Leftrightarrow x_1,...,x_n$ seperable over $K$

• $x\in L\Rightarrow$ the minimal polynomial $x^n+a_{n-1}x^{n-1}+...+a_0$ of $x$ over $F$ is seperable $\Rightarrow K[a_0,...,a_{n-1},x]/K[a_0,...,a_{n-1}]$ and $K[a_0,...,a_{n-1}]/K$ seperable.

First arrow: $L/F$ is seperable

Second arrow: $K[x_1,...,x_n]/K$ seperable $\Leftrightarrow x_1,...,x_n$ seperable over $K$

2017-12-03 04:26:23