Help me Understanding Non Abelian Kummer Extensions

Question

[![I am doing my undergraduate research work on the Non-Abelian Kummer Extensions. I am following the book "Algebra" by Serge Lange. I have understood the difference between the abelian Kummer Extensions and the non-abelian Kummer Extensions, which is that the earlier one requires the presence of a primitive root of unity in the base field, while the latter one requires no such conditions, and hence its Galois group may be non-Abelian. The difficulty I am facing is with the example given in section 11 of chapter 3. The image of that example is attached. Can someone kindly help me understand the basic idea of this example and the notations used at the end of the example, and please guide me to some help material or literature about non-abelian Kummer Extensions. Thank you!]$$ $$ [Non-Abelian Kummer Extensions] $$ $$ We are interested in the splitting field of the equations $x^{n}-a=0$ when the $n^{th}$ roots of unity are not contained in the ground field. More generally, we want to know the Galois group of simultaneous equations of this type. For this purpose, we axiomatize the pattern of proof to an additive notation, which in fact makes it easier to see what is going on. We fix an integer N>1, and we let M range over positive integers dividing N. We let P be the set of primes dividing N. We let G be a group and let:$$ $$ A=G-module such that the isotropy group of any element of A is of finite index in G. We also assume that A is divisible by the primes $p|N$, that is $pA=A\ \forall p\in{P}$. $$\ $$ $\Gamma=$ finitely generated subgroup of A such that $\Gamma$ is pointwise fixed by G. $$ $$ We assume that $A_{n}$ is finite. Then $\frac{1}{n}\Gamma$ is also finitely generated. Note that $A_{n}\subset{\frac{1}{n}\Gamma}$ $$ $$ Example: For our purpose here, the above situation summarizes the properties which hold in the following suituation. Let $K$ be finitely generated field over the ratiopnal number, or even a finite extension of the rational numbers. We let $A$ be the multiplicative group of the algebraic closure $K^{a}$. We let $G=G_{K}$ be the Galois group $Gal(K^{a}/K)$. We let $\Gamma$ be a finitely generated subgroup of the mutiplicative group $K^{*}$. Then all the above properties are satisfied. We see that $A_{n}=\mu_{n}$ is the group of $N$-th roots of unity. The group written $\frac{1}{N}\Gamma$ in additive notation is written $\Gamma^{1/N}$ in multiplicative notation.$$ $$ I am facing trouble in understanding the next part, which states that$$ $$ Next we define the appropriate groups of analogous to the Galois groups of the Kummer theory as follows. For any $G$-submodule $B$ of $A$, we let: $$ $$ $G(B)$=image of $G$ in $Aut(B)$. $$ $$ $G(N)=G(A_{N})$=image of G in $Aut(A_{N})$ $$ $$ $H(N)$=subgroup of G leaving $A_{n}$ pointwise fixed, $$ $$ $H_{\Gamma}(M,N)$(for M|N)=image of $H(N)$ in $Aut(\frac{1}{M}\Gamma)$ $$ $$ Thus we have an exact sequence $$ $$ $0\to{H_{\Gamma}(M,N)}\to{G(\frac{1}{M}\Gamma+A_{n})}\to{G(N)}\to{0}$ $$ $$ I am totally stuck at the end of this example. Can someone kindly explain it to me. And please guide me to some help material or literature about non-abelian Kummer Extensions. Thank you!