query: If $F\subseteq L\subseteq K$ are fields with $K/L$ and $L/F$ Galois, then $K/F$ is Galois?.
positive_passages.docid: 27440
positive_passages.text: Is normal extension of normal extension always normal?
negative_passages.docid: 56439
negative_passages.text: $f \in K[X]$ of degree $n$ with Galois group $S_n$, why are there no non-trivial intermediate fields of $K \subset K(a)$ with $a$ a root of $f$?
negative_passages.docid: 112556
negative_passages.text: When is $F(\sqrt c)$ a field?
negative_passages.docid: 109034
negative_passages.text: Does a finite $[E:K]_s$ imply that $[E_s:K]$ is finite?
negative_passages.docid: 19202
negative_passages.text: $\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$ if and only if $d$ divides $n$
negative_passages.docid: 143219
negative_passages.text: If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$
negative_passages.docid: 100439
negative_passages.text: Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$
negative_passages.docid: 86877
negative_passages.text: Let $F$ be a Galois extension over $\mathbb{Q}$ with $[F:\mathbb{Q}]=2^n$, then all elements in $F$ are constructible
negative_passages.docid: 149849
negative_passages.text: Let $F$ be a field and $f: \mathbb{Z} \to F$ be a ring epimorphism.
negative_passages.docid: 58424
negative_passages.text: If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field.
negative_passages.docid: 41815
negative_passages.text: Galois extension of intersection of fields
negative_passages.docid: 9129
negative_passages.text: Show that $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ and $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}))\cong \mathbb{Z_n}^*$
negative_passages.docid: 133561
negative_passages.text: show that if $K$ is a field then $K[x]$ is principal
negative_passages.docid: 137715
negative_passages.text: If $[L:K]$ is prime, show that $L/K$ is a simple extension.
negative_passages.docid: 45219
negative_passages.text: Proving that $C$ is a subset of $f^{-1}[f(C)]$
negative_passages.docid: 119305
negative_passages.text: Is the splitting field equal to the quotient $k[x]/(f(x))$ for finite fields?
negative_passages.docid: 49768
negative_passages.text: If Gal(K,Q) is abelian then |Gal(K,Q)|=n