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

Step 5 of 8: Galois's criterion: solvability of the equation equals solvability of the group

In plain words

Steps 1–4 built a one-way bridge: a radical formula for $f$ produces a solvable Galois group. Galois's real insight was that the bridge also runs the other way — from a solvable group one can always manufacture radical formulas, by essentially reversing the construction (the classical technique for this uses Lagrange resolvents, sums of roots weighted by roots of unity). So the two notions, one about equations and one about groups, turn out to be exactly the same fact seen from two sides.

f \text{ solvable by radicals} \iff \mathrm{Gal}(f/\mathbb{Q}) \text{ is a solvable group}

Detailed analysis

Combining Steps 1–4 gives one direction: if $f$ is solvable by radicals, its Galois group is solvable. Galois proved the converse as well — from a solvable Galois group one can reconstruct a radical tower whose top field contains all the roots — so the two conditions coincide exactly, giving Galois's criterion: $f$ is solvable by radicals if and only if $\mathrm{Gal}(f/\mathbb{Q})$ is a solvable group. This criterion, presented by Galois around 1830–1832 and published posthumously in 1846, turns a question about formulas into a purely finite question about the structure of a permutation group.

Knowledge used in this step

Galois theory Group theory Fundamental theorem of Galois theory