When do intermediate fields being Galois imply entire extension is Galois? Let $k \subset K$ be Galois. The Fundamental Theorem of Galois Theory gives us a criteria for when any intermediate field $k \subset E \subset K$ is Galois over our base field $k$. Also, it is in general not true that if $k \subset E$ and $E \subset K$ are both Galois, then $k \subset K$ will be Galois. For example, take $k = \mathbb{Q}, E = \mathbb{Q}[\sqrt{2}]$, and $K = \mathbb{Q}[\sqrt[4]{2}]$.
However, are there any minimal restrictions we can put on on the extensions $k \subset E \subset K$ such that when $E$ is Galois over $k$ and $K$ is Galois over $E$ that guarantees us $K$ being Galois over $k$?
 A: The terms of your question are too vague. In the tower of (finite) extensions $k < E < K $, with $E/k$ and $K/E$ galois, you need a minimal amount of information relative to the two intermediate extensions. Suppose e.g. that $G=Gal(E/k)$ is given, and $K$ is determined by a set of generators over $E$, say $K=E(A), A$ finite $\subset K$. Assume that separability is automatic. Then, checking normality, it is straightforward to see that $K/k$ is galois iff for all $\sigma \in G$, all the prolongations $\tau$ of $\sigma$ to $k$-embeddings of $E$ into a separable closure of $k$ stabilize $E$, i.e. $\tau (A) \subset E$. The problem is thus reduced to a convenient expression of this last property.
A simple instructive illustration comes from Kummer's theory. Suppose e.g. that $k$ has characteristic $0$ and contains the group $\mu_n$ of $n$-th roots of unity, and moreover that $H=Gal(K/E)$ is abelian of exponent $n$. By Kummer theory, there exists a finite subgroup $B$ of $E^*/{E^*}^n$, the Kummer radical of $K/E$, such that $B\cong Hom(H,\mu_n)$ and $K=E(B^{1/n})$ in the usual notations. The criterion for the normality of $K/k$ is then the $G$-stability of $B$. Things become even more visible when, in addition, $n$ is a prime $p$, $G$ is a cyclic $p$-group of order $p^n$ and $B$ is cyclic of order $p$, in which case $G$ acts trivially on $B$. If $Gal(K/k)$ is abelian, it's isomorphic to either  $C_p \times C_{p^n}$ or $C_{p^{n+1}}$; in general, $Gal(K/k)$ is characterized up to isomorphism by a cohomology class of $H^2(G, C_p)$; an istructive example is when $Gal(K/k)$ has order $p^3$. Other analogous variations can be considered in the kummerian situation. In the case of number fields, with $K/E$ abelian, a $G$-module description of $H$ in terms of class-field theory can be used, but of course things get more complicated.
