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