Galois group of Solvable extension is solvable. (Proof) I gonna show that if $$F\subset L$$ is solvable and Galois the theorem above holds.

Suppose $$F\subset L$$ is Galois and radical.

Since L is radical over F, we have $$F\subset F_1\subset F_2 ...\subset F_n=L$$ where $$F_{i+1} =F_i(\gamma_i)$$ and $$\gamma_i^{m_i}\in F_i$$ for some $$m_i >0$$

Each $$c_i$$ is the $$m_i^{th}$$ root of unity.

We can adjoin the $$m_i^{th}$$ roots of unity to each subfield above to show that $$F(c_1,c_2,...,c_n)\subset L(c_1,c_2,...,c_n)$$ is radical.

i.e. $$F(c_1,c_2,...,c_n)\subset F_1(c_1,c_2,...,c_n)\subset F_2(c_1,c_2,...,c_n) ...\subset F_n(c_1,c_2,...,c_n)=L(c_1,c_2,...,c_n)$$

where $$F_{i+1}(c_1,c_2,...,c_n) =F_i(c_1,c_2,...,c_n)(\gamma_i)$$ and $$\gamma_i^{m_i}\in F_i(c_1,c_2,...,c_n)$$ for some $$m_i >0$$

Note that $$F_n(c_1,c_2,...,c_n)$$ is Galois over $$F(c_1,c_2,...,c_n)$$

Let $$K_i= F_i(c_1,c_2,...,c_n)$$

a) We need to show that $$K_{i-1}\subset K_i$$ is Galois and its Galois group is cyclic.

Since $$\gamma_i^{m_i}\in F_{i-1}\subset K_{i-1}$$ and $$c_i\in K_{i-1}$$ we have $$x^{m_i}-\gamma_i^{m_i}\in K_{i-1}[x]$$ with distinct roots $$\gamma_i,\gamma_i c_i,...,\gamma_i c_i^{m_i-1}$$

Hence the splitting field of $$x^{m_i}-\gamma_i^{m_i}$$ is $$K_{i-1}(\gamma_i)=K_i$$. Hence, the Galois assertion is proved.

$$\forall \sigma, \tau \in Gal(K_i/K_{I-1})$$ $$\sigma(\gamma_i)=\gamma_i c_i^a$$ and $$\tau(\gamma_i)=\gamma_i c_i^b$$ where $$1\leq a,b\leq m_i$$.

Hence, $$\sigma\tau(\gamma_i)=\gamma_i c_i^{b+a}=\tau\sigma(\gamma_i)$$

Therefore, abelian assertion is proved.

b) We now have the following chain:

$$Gal(K_n/k_n)\subset .... \subset Gal(K_n/K_0)$$

Let us consider $$K_{i-1}\subset K_i \subset K_n$$. Note that $$K_n$$ is Galois over $$K_{i-1}$$.

From (a) we have that $$Gal(K_i/K_{i-1})$$ is cyclic hence, $$Gal(K_n/K_{i-1})/Gal(K_n/K_i)$$ is cyclic and we have $$Gal(K_n/K_{i-1})$$ is normal in $$Gal(K_n/K_i)$$

Therefore $$Gal(K_n/K_0)$$ is radical which implies that Gal(L/F) is radical.

Are there any flaws or loopholes?