TITLE: An exercise on cyclic extensions of Hungerford's book, Algebra.
QUESTION [9 upvotes]: I'm trying to show the following exercise of the book, it is in the section of cyclic extensions and says the following:

Let $\overline{\mathbb Q}$ be a fixed algebraic closure of $\mathbb Q$, let $v\in\overline{\mathbb Q}-\mathbb Q$ and let $E\subseteq\overline{\mathbb Q}$ be a subfield maximal with respect to the property $v\notin E$. Prove that every finite dimensional extension of $E$ is cyclic.

Well, this is my attempt of proof:
First I prove that one can restrict himself to a finite Galois extensions. Then I prove that $E(v)$ is a Galois extension of $E$, by noting that for every $\sigma\in Aut_E E(v)$, $\sigma(v)\notin E$; and that $E(v)$ is a cyclic extension of $E$, because the absence of intermediate fields and applying the fundamental theorem of Galois theory . Next, I intend to proceed inductively in $n=[F:E]$ with the case $n=1$ being trivial:
suppose that the proposition is true for every $m\leq r$, and that $r+1=[F:E]$; consider the intermediate field $E\subseteq E(v)\subseteq F$, if $E(v)=F$ the proposition has already been proven. If not then $[F:E(v)]\leq r$, and we have that $F$ is Galois over $E(v)$ (fundamental theorem), therefore the induction hypothesis shows that $Aut_{E(v)}F$ is cyclic. The fact that $E(v)$ is Galois over $E$ implies that $Aut_{E(v)}F$ is normal in $Aut_E F$ and again by the fundamental theorem $Aut_E E(v)=Aut_E F/Aut_{E(v)}F$.
At this point I had been thinking about a proposition of the form "If a finite group G contains a normal cyclic subgroup such that the quotient $G/N$ is also cyclic then G is cyclic"; which is false, even if one replace normal cyclic, for maximal normal cyclic, and things like this.
I don't know what to do at this stage, I would be very grateful for a solution that follows this path. Thanks in advance.

REPLY [6 votes]: Hints:

Let $F$ be any finite extension of $E$. Show that $F/E$ is separable, simple and that there exists an extension field $M/E$ such that $F\subseteq M$, and $M/E$ is a finite Galois extension.
Show that $H=\operatorname{Gal}(M/E(v))$ is the unique maximal subgroup of $G=\operatorname{Gal}(M/E)$.
Show that any element $\sigma\in G\setminus H$ generates $G$.
Show that $F/E$ is cyclic.