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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 $E \supset L \supset F$

  1. $E \supset L$ is algebraic and $L \supset F$ is algebraic $\Rightarrow E \supset F$ is algebraic
  2. $E \supset F$ is algebraic $\Rightarrow E \supset L$ is algebraic and $L \supset F$ is algebraic
  3. $E \supset L$ is separable and $L \supset F$ is separable $\Rightarrow E \supset F$ is separable
  4. $E \supset F$ is separable $\Rightarrow E \supset L$ is separable and $L \supset F$ is separable
  5. $E \supset L$ is splitting and $L \supset F$ is splitting $\Rightarrow E \supset F$ is splitting
  6. $E \supset F$ 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

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