Worked solution: Abel–Ruffini theorem and the Galois solvability criterion (1824)

Step 3 of 8: The Galois correspondence turns fields into groups

In plain words

The Galois correspondence is like a perfect dictionary between two languages: every intermediate field sitting between $F$ and the big field $E$ has an exact translation as a subgroup of automorphisms, and bigger fields translate to smaller groups (and vice versa). Once the tower of fields from Step 2 is translated this way, the whole radical tower becomes a tower of groups sitting inside $\mathrm{Gal}(E/F)$ .

E/F \text{ normal}, \qquad \{\text{subfields } K,\ F\subseteq K\subseteq E\} \;\longleftrightarrow\; \{\text{subgroups } H \le \mathrm{Gal}(E/F)\}

Detailed analysis

For a normal (Galois) extension $E/F$ , there is a one-to-one, inclusion-reversing correspondence between the intermediate fields $K$ with $F\subseteq K\subseteq E$ and the subgroups $H$ of $G=\mathrm{Gal}(E/F)$ : the field $K$ corresponds to the subgroup $\mathrm{Gal}(E/K)$ of automorphisms fixing $K$ pointwise, and a subgroup $H$ corresponds to its fixed field. Applying this correspondence to the refined radical tower from Step 2 converts the chain of fields $F=K_0'\subseteq K_1'\subseteq\cdots\subseteq K_m'=E$ into a chain of subgroups $G=H_0 \ge H_1 \ge \cdots \ge H_m=1$ , each $H_i$ normal in $H_{i-1}$ with cyclic quotient $H_{i-1}/H_i \cong \mathrm{Gal}(K_i'/K_{i-1}')$ .

Terms in this step

Galois correspondence: The dictionary, for a normal extension, that pairs each intermediate field with a subgroup of the Galois group, in an inclusion-reversing way.
Normal subgroup: A subgroup $N$ of $G$ that is unchanged by conjugation ( $gNg^{-1}=N$ for every $g\in G$ ), which is exactly what lets one form the quotient group $G/N$ .

Knowledge used in this step

Galois theory Group theory Fundamental theorem of Galois theory