# Characterization of solvable normal extention over a field

I think I proved the following result. Is this correct? Is there any reference?

Definition
A finite extention of a field $E/K$ is called a general radical tower when there is a sequence of extention fields $K_i$, for $i = 0, 1, \ldots, m$ which satisfies the following conditions:

1. $K = K_0$, $E = K_m$

2. For each $i = 0, \ldots, m-1$, $K_{i+1} = K_i(b)$, where $b^q = a ∈ K_i$ and $q$ is a prime which may or may not be the characteristic of $K$ and $X^q - a$ is an irreducible polynomial in $K_i[X]$ or $K_{i+1}/K_i$ is a Galois extention of degree $p$, where $p$ is the characteristic of $K$.

Proposition
Let $L$ be a finite (not necessarily separable) normal extention of a field $K$. Let $G$ be the automorphism group of $L/K$. Then the following conditions are equivalent.

1. $G$ is solvable.

2. There is a general radical tower $E/K$ such that $L$ is contained in $E$.

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Magic words are "Kummer theory". –  Arturo Magidin Apr 15 '12 at 3:31
But Kummer theory does not treat Artin-Schreier extensions nor inseparable extentions. –  Makoto Kato Apr 15 '12 at 4:23
I crossposted this question in MO. I didn't get any answer nor any comment there so far. –  Makoto Kato Apr 24 '12 at 8:46