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

Step 1 of 8: Radical towers formalize 'solvable by radicals'

In plain words

A radical formula, like the quadratic formula $x=(-b\pm\sqrt{b^2-4ac})/2a$ , is really just a recipe of finitely many additions, multiplications, divisions and root extractions applied to the coefficients. Writing that recipe down as a sequence of fields — start with $\mathbb{Q}$ , throw in one new radical at a time — turns a vague hope ("is there a formula?") into a precise mathematical object one can actually study.

\mathbb{Q} = K_0 \subset K_1 \subset \cdots \subset K_r, \quad K_i = K_{i-1}(\sqrt[n_i]{a_i})

Detailed analysis

A polynomial equation $f(x)=0$ is solvable by radicals when its roots lie in some field $K_r$ obtained from $\mathbb{Q}$ by a finite chain of radical extensions $K_i = K_{i-1}(\sqrt[n_i]{a_i})$ , each adjoining an $n_i$ -th root of an element already in $K_{i-1}$ .

Terms in this step

Field: A set of numbers, such as $\mathbb{Q}$ or $\mathbb{R}$ , closed under addition, subtraction, multiplication, and division by anything nonzero.
Field extension: A field $L$ containing a smaller field $K$ , written $L/K$ ; new elements are built from $K$ by adjoining extra numbers such as roots.
Radical extension: A one-step field extension $K_{i-1}(\sqrt[n]{a})$ formed by adjoining an $n$ -th root of some element $a$ already present in $K_{i-1}$ .

Knowledge used in this step

Galois theory Group theory