I'm putting together an outline for a paper on Galois theory. There are a few equivalent definitions of a solvable group, and I need to make sure that the one I'd like to use works, or more specifically, that the following is correct:
Let $f$ be a polynomial with coefficients in $K$, and let $K'$ be the field obtained by adjoining the roots of $f$ to $K$. Then there is a sequence of sub-groups: $$\text{Gal}(K'/K)=G_1\triangleright G_2\triangleright\ ...\ \triangleright G_n=1$$ where for each $i$, $[G_i:G_{i+1}]$ is prime.
Is this too strong a statement?
Wikipedia states, "for finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order." However, this seems stronger than my statement, in which I don't require $(G_i)$ to be the composition series.
The condition that $M\lhd G$ is superfluous: if $M\leqslant N\leqslant G$ then $[G:M]=[G:N][N:M]$ so either $N=M$ or $G=N$, i.e. $M$ is maximal.