Given a field $F$ and polynomial $f \in F[x]$, the splitting field $K/F$ is the smallest extension in which $f$ factors completely into linear factors. The Galois group $\mathrm{Gal}(K/F)$ consists of field automorphisms of $K$ that fix $F$ pointwise.

The fundamental theorem of Galois theory: for finite Galois extensions, there is an order-reversing bijection between intermediate fields $F \subseteq L \subseteq K$ and subgroups $H \leq G = \mathrm{Gal}(K/F)$, sending $L \mapsto \mathrm{Gal}(K/L)$.

A polynomial is solvable by radicals iff its Galois group is solvable (composition series with abelian factors). The Galois group of a generic degree-$n$ polynomial is $S_n$. $S_n$ is solvable for $n \leq 4$ — hence quadratic / cubic / quartic formulas exist. For $n \geq 5$, $A_n$ is simple non-abelian, so $S_n$ is not solvable — Abel–Ruffini: no general solution by radicals exists for the quintic and higher.

Interactive: the 3 cube roots of $a$ and their Galois action

The Galois group of $x^3 - a$ over $\mathbb Q$ (with $a$ not a cube) is $S_3$, generated by complex conjugation and a rotation. The roots sit on a circle in $\mathbb C$.