Galois extentions 

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*If $F \subset K \subset E$, with $E/K$ and $K/F$ galois, then $E/F$ is galois.


*If $F \subset K \subset E$, with $E/F$ galois, then $E/K$ and $K/F$ are galois.
$\def\less{\smallsetminus}$

For the first one: $E/F$ is galois because $E/K$ is galois you can extend an automorphism $\delta$ on $F$ to an automorphism $δ'$ on $K$ such that $δ'(α) \neq α$  for $α \in K \less F$. And in a similar way for elements in $E \less K$. Therefore there is an automorphism on $E$ which permutes all elements not in $F$.
How would I prove the second one is right or wrong?

If $Gal(E/F)$ is abelian, then the second statement is true.
In general case, $E/K$ is Galois, given $E/F$ is. This is the statement of Fundamental Theorem of Galois theory.
However $K/F$ needn't be Galois if $Gal(E/F)$ is not abelian.
Let $E$ be the splitting field of $x^{3} - 2$, an irreducible polynomial, over $\mathbb{Q} = F$ and let $K =  \mathbb{Q} (2^{1/3})$. Then $E$ is Galois over $F$ , because it's the splitting field of a polynomial,hence over $K$ as well,  However $K/F$ is not a normal extension (Why?)  , hence not Galois. 
