With the idea of the group of an equation, Galois answered the question left open after Abel: which equations can be solved by radicals? The answer does not depend on any trick. It depends only on the shape of the group.

Each root cuts the group

Solving an equation by radicals means adding roots, one at a time, to the numbers already known. At the start we know only the rational numbers, and the group of the equation is whole. Each time we add a root, such as $\sqrt3$ or $\sqrt[3]{5}$, new true relations appear, and fewer permutations respect them. The group shrinks. When the group becomes just the identity, all the roots of the equation are written with known numbers: the equation is solved.

Galois showed that each root of prime index $p$, a square, cube or fifth root, shrinks the group in a very particular way: the new group is a normal subgroup of the previous one, and fits inside it exactly $p$ times. The conclusion is Galois’s theorem:

$$\boxed{\text{The equation can be solved by radicals}\iff\text{the group can be cut in normal steps of prime index down to the identity.}}$$

A group that can be cut in this way is called, in honour of this story, solvable.

Degree 3: Cardano’s formula

The group of the general equation of degree 3 is that of the six permutations of three roots, which is the same as that of the six symmetries of the triangle. It can be cut like this:

$$\underbrace{\{e,r,r^2,s_1,s_2,s_3\}}_{6}\ \supset\ \underbrace{\{e,r,r^2\}}_{3}\ \supset\ \underbrace{\{e\}}_{1}.$$

The indices are $6\div3=2$ and $3\div1=3$. This means: first a square root, then a cube root. That is exactly the shape of Cardano’s formula, in which a square root appears inside a cube root:

$$x=\sqrt[3]{-\frac q2+\sqrt{\frac{q^2}4+\frac{p^3}{27}}}+\sqrt[3]{-\frac q2-\sqrt{\frac{q^2}4+\frac{p^3}{27}}}.$$

The group explains the formula before it is written.

Degree 4: Ferrari’s formula

The 24 permutations of four roots can be cut in four steps:

$$24\ \supset\ 12\ \supset\ 4\ \supset\ 2\ \supset\ 1,$$

with indices $2, 3, 2, 2$, all prime. Ferrari’s formula uses, in this order, a square root, a cube root (to solve Lagrange’s auxiliary cubic) and two more square roots.

Degree 5: the group that cannot be cut

The $120$ permutations of five roots can be cut once, in half, into a normal subgroup of $60$ elements, the so-called even permutations. But there the chain stops: this group of 60 elements has no normal subgroup other than $\{e\}$ and itself. To reach the identity, a step of index 60 would be needed, and 60 is not prime. It is impossible.

That is why the general equation of the fifth degree has no formula: it is Abel’s theorem, seen now through Galois’s eyes. But Galois’s theory goes further. For each particular equation, it is enough to compute the group. The equation $x^5-2=0$ has a group of 20 elements, which is solvable, and indeed it can be solved: $x=\sqrt[5]2$. On the other hand $x^5-x-1=0$ has the group of all 120 permutations, and its real root, $1.1673\ldots$, cannot be written with radicals.

Exercise: why can the chain of the triangle not start with the subgroup {e, s₁}?

Because it is not normal. Take $g=r$ and $h=s_1$ and compute “undo $g$, do $h$, redo $g$”, that is, $r^2\star s_1\star r$. From the table, $r^2\star s_1=s_3$, and $s_3\star r=s_2$. The result, $s_2$, is not in $\{e,s_1\}$. For the rotations the computation always gives a rotation, which is why the chain goes through $\{e,r,r^2\}$.

Exercise: the group of x⁵ − 2 = 0 has 20 elements and can be cut as 20 ⊃ 10 ⊃ 5 ⊃ 1. Which roots does the formula use?

The indices are $20\div10=2$, $10\div5=2$ and $5\div1=5$: two square roots and a fifth root. The two square roots are used to write $\omega=\cos72°+i\sin72°$, because $\cos72°=\frac{\sqrt5-1}4$. The fifth root is $\sqrt[5]2$. The five roots of the equation are $\sqrt[5]2,\ \sqrt[5]2\,\omega,\ \sqrt[5]2\,\omega^2,\ \sqrt[5]2\,\omega^3$ and $\sqrt[5]2\,\omega^4$.

Sources: Évariste Galois, “Mémoire sur les conditions de résolubilité des équations par radicaux” (1831), in Journal de mathématiques pures et appliquées 11 (1846) · Felix Klein, Vorlesungen über das Ikosaeder (1884) · Ian Stewart, Galois Theory (CRC Press, 4th ed., 2015).

All topics on Évariste Galois · ← What a group is: the six symmetries of the triangle · Galois imaginaries: finite fields →

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