To jump with both feet over the calculations; to group the operations, to classify them according to their difficulties and not according to their forms: such is, in my view, the mission of future geometers.

— Évariste Galois, preface to a manuscript (1831)

Sixty years before Galois, Lagrange had looked at expressions made from the roots of an equation and counted how many values they take when the roots change places. Galois went one step further. Instead of looking at the expressions, he looked at the swaps. And not at all of them: only at those the equation allows.

The allowed swaps

Take an equation with rational coefficients and call its roots $a, b, c,\dots$ Certain relations hold between the roots: equalities with rational coefficients that are true, such as $a+b=0$ or $a^2=2$. Galois defined the group of the equation as the set of permutations of the roots that turn every true relation into another true relation.

The simplest example is $x^2-2=0$, with roots $a=\sqrt2$ and $b=-\sqrt2$. The following hold, for instance:

$$a+b=0,\qquad ab=-2,\qquad a^2=2,\qquad b^2=2.$$

Swapping $a$ with $b$, all four remain true. The group has two elements: swapping nothing and swapping the two roots.

Compare with $x^2-3x+2=0$, with roots $a=1$ and $b=2$. Here the relation $a=1$ holds. Swapping the roots turns it into $b=1$, which is false. So the swap is not allowed, and the group contains only the identity. This is no accident:

A rational expression in the roots is a rational number exactly when no permutation of the group changes it.

This is the central principle of the theory. In particular, the roots are all rational exactly when the group is just the identity. The larger the group, the more “hidden” the roots are.

Four roots, four swaps

The equation $x^4-5x^2+6=0$ factors as $(x^2-2)(x^2-3)=0$. The roots are

$$a=\sqrt2,\qquad b=-\sqrt2,\qquad c=\sqrt3,\qquad d=-\sqrt3.$$

There are $4!=24$ ways of shuffling four roots, but most of them break some relation. Swapping $a$ with $c$, for example, turns the true relation $a^2=2$ into $c^2=2$, which is false. Only the swaps that send $\sqrt2$ to $\pm\sqrt2$ and $\sqrt3$ to $\pm\sqrt3$ remain:

Permutation $a$ goes to $c$ goes to
identity $a$ $c$
swap $a\leftrightarrow b$ $b$ $c$
swap $c\leftrightarrow d$ $a$ $d$
both swaps $b$ $d$

The group has four elements, out of 24 possible. The equation, which seemed to be of degree 4, is really solved with two independent square roots, and the group shows it.

Three roots, six swaps

The equation $x^3-2=0$ has one real root, $\alpha=\sqrt[3]2$, and two complex ones. With $\omega=\frac{-1+i\sqrt3}2$, which satisfies $\omega^3=1$, the roots are

$$\alpha,\qquad\alpha\omega,\qquad\alpha\omega^2.$$

In this case all $3!=6$ permutations are allowed, and the group is the largest possible for three roots. It is the same group as the symmetries of an equilateral triangle, and not by chance: the three roots, drawn in the complex plane, form an equilateral triangle centred at the origin.

Exercise: what is the group of x² − 4x + 1 = 0?

The roots are $a=2+\sqrt3$ and $b=2-\sqrt3$. The relations $a+b=4$ and $ab=1$ do not change when $a$ and $b$ are swapped. Any true relation between $a$ and $b$ stays true when $\sqrt3$ is replaced by $-\sqrt3$, so the swap is allowed. The group has two elements. Since $a$ is not rational, the group cannot be just the identity, which confirms the calculation.

Exercise: in the equation x⁴ − 5x² + 6 = 0, is the expression a + c a rational number? And a + b?

$a+c=\sqrt2+\sqrt3$. The swap $c\leftrightarrow d$, which is in the group, turns it into $a+d=\sqrt2-\sqrt3$, a different value. So $a+c$ is not rational. On the other hand $a+b=0$ is not changed by any of the four permutations: it is rational, as it should be.

Sources: Évariste Galois, “Mémoire sur les conditions de résolubilité des équations par radicaux” (1831), published by Liouville in the Journal de mathématiques pures et appliquées 11 (1846) · Peter M. Neumann, The Mathematical Writings of Évariste Galois (2011) · Ian Stewart, Galois Theory (CRC Press, 4th ed., 2015).

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