I just took my algebra final, and I had a few questions about intermediate fields and the properties of their extensions
True or False...For $E \supset L \supset F$
- $E \supset L$ is algebraic and $L \supset F$ is algebraic $\Rightarrow E \supset F$ is algebraic
- $E \supset F$ is algebraic $\Rightarrow E \supset L$ is algebraic and $L \supset F$ is algebraic
- $E \supset L$ is separable and $L \supset F$ is separable $\Rightarrow E \supset F$ is separable
- $E \supset F$ is separable $\Rightarrow E \supset L$ is separable and $L \supset F$ is separable
- $E \supset L$ is splitting and $L \supset F$ is splitting $\Rightarrow E \supset F$ is splitting
- $E \supset F$ is splitting $\Rightarrow E \supset L$ is splitting and $L \supset F$ is splitting
My guess is that 1,3,5 are true, but I can't work out a proof for 2,4, or 6. My guess is that they are false.
Could anyone point me in the right direction? Thanks in advance