First, I define a radical tower as I have it in front of me (in case our definitions differ):
By a radical tower over a field F we mean a sequence of finite extensions $F=F_0{\subset}F_1{\subset}..{\subset}F_r$ having the property that there exist positive integers $d_i$, elements $a_i$ in $F_i$ and $\alpha_i$ with ${\alpha_i}^{d_i}=a_i$ such that $F_{i+1}=F_i(\alpha_i)$. We say that E is contained in a radical tower if there exists a radical tower above such that $E{\subset}F_r$.
Anyway, now onto the main problem: Let E be a finite extension of F (F characteristic 0) and suppose E is contained in a radical tower. Show that there exists a radical tower $F{\subset}E_0{\subset}E_1{\subset}...{\subset}E_m$ such that:
a) $E_m$ is Galois over F and $E{\subset}E_m$
b) $E_0=F(\zeta)$ where $\zeta$ is a primitive nth root of unity
c) For each i, $E_{i+1}=E_i(\alpha_i)$ where ${\alpha_i}^{d_i}=a_i$ $\in$$E_i$, and $d_i$|n.
I am rather stuck on this problem and have not been able to make much headway. Would anyone be able to give any hints/help me solve this problem?
Thank you
So, we have a radical tower $F=F_0\subset F_1\subset\cdots\subset F_r$ with $F_{i+1}=F_i(\alpha_i)$, $\alpha_i^{d_i}=a_i$, $a_i$ in $F_i$, and $E\subset F_r$. Let $n$ be the least common multiple of $d_1,\dots,d_r$. Now consider the tower $F\subset E_0\subset E_1\subset\cdots\subset E_m$ given by $E_0=F(\zeta)$ where $\zeta$ is a primitive $n$th root of unity, and $E_{i+1}=E_i(\alpha_i)$.
This is clearly a radical tower.
Condition b) is clearly met.
Condition c) is met since $n$ is the least common multiple of the $d_i$.
$E$ is contained in $F_r$, and $F_r$ is contained in $E_m$, so $E\subset E_m$.
So all that remains is to show that $E_m/F$ is Galois.
Now $E_m=F(\zeta,\alpha_1,\dots,\alpha_r)$, so $E_m/F$ is finite. It is also normal, since it contains all the conjugates of all of its generators. So, it's Galois.