In 1770 algebra had a problem that had been stuck for more than two centuries. There were formulas for solving equations of degree 2, the quadratic formula, and also of degrees 3 and 4, published by Cardano in 1545. For degree 5, nobody could find anything. The best mathematicians had tried, and every method that worked up to degree 4 jammed at 5.

Lagrange did not try yet another method. He asked a different question: why do the old methods work?

I propose in this memoir to examine the various methods found so far for the algebraic solution of equations, to reduce them to general principles, and to show a priori why these methods succeed for the third and fourth degrees and fail for higher degrees.

— Joseph-Louis Lagrange, Reflections on the Algebraic Solution of Equations (1771)

The memoir runs to more than 200 pages, published by the Berlin Academy in 1770 and 1771.

Shuffling the roots

The central idea is to look at expressions built from the roots of the equation and see what happens to them when the roots swap places.

Start with degree 2. The equation $x^2+bx+c=0$ has roots $x_1$ and $x_2$, and it has been known since Viète that

$$x_1+x_2=-b,\qquad x_1x_2=c.$$

These two expressions do not change if we swap $x_1$ with $x_2$. That is why they can be written in terms of the coefficients. The difference $x_1-x_2$, on the other hand, does change: it becomes $x_2-x_1$, its opposite. It takes two values. But its square goes back to taking only one, and so it can be written in terms of the coefficients:

$$(x_1-x_2)^2=(x_1+x_2)^2-4x_1x_2=b^2-4c.$$

Now just take the root: $x_1-x_2=\pm\sqrt{b^2-4c}$. Knowing the sum and the difference, the roots come out at once, and the result is the quadratic formula. The $\pm$ in the formula is exactly the sign of the two values that $x_1-x_2$ can take.

The secret of degree 4

Lagrange repeated the same analysis for degrees 3 and 4. In degree 4, with roots $x_1, x_2, x_3, x_4$, Ferrari’s trick amounts to looking at the expression

$$t=x_1x_2+x_3x_4.$$

There are $4!=24$ ways of shuffling four roots, but $t$ takes only three values:

$$\begin{gathered}x_1x_2+x_3x_4,\\ x_1x_3+x_2x_4,\\ x_1x_4+x_2x_3.\end{gathered}$$

These three values are the roots of an equation of degree 3, whose coefficients can be computed from those of the original equation. Since degree 3 can already be solved, we find $t$, and from it the roots. The degree-4 equation can be solved because there is an expression in the roots that takes fewer than four values.

For degree 5, Lagrange looked for an expression in the five roots that took few values, and did not find one. He suspected the problem had no solution, but could not prove it. The Italian Paolo Ruffini gave an almost complete proof in 1799 that no general formula exists, and the Norwegian Niels Abel settled the question in 1824. A few years later Évariste Galois turned Lagrange’s idea into an entire theory, group theory.

The theorem

In the middle of this analysis, Lagrange proved a result that now bears his name:

If an expression in $n$ letters is shuffled in all $n!$ possible ways, the number of different values it takes divides $n!$.

For example, the four-letter expression $t$ takes 3 values, and 3 divides 24. The expression $x_1-x_2$, in two letters, takes 2 values, and 2 divides $2!=2$.

Why does it divide? The swaps that do not change the expression form a group of, say, $k$ shufflings. Each different value is obtained by exactly $k$ of the $n!$ shufflings. So

$$n!=k\times(\text{number of values}),$$

and the number of values is a divisor of $n!$. For $t$, the swaps that leave it unchanged number 8, and $24=8\times3$.

In modern language, which only appeared in the nineteenth century, Lagrange’s theorem says: in a finite group, the number of elements of a subgroup divides the number of elements of the group. It is one of the first theorems every algebra student learns. Lagrange did not use the word group, but the whole idea was there.

Exercise: how many values does x₁ + x₂ take when four roots are shuffled?

The values are the sums of two different roots: $x_1+x_2$, $x_1+x_3$, $x_1+x_4$, $x_2+x_3$, $x_2+x_4$ and $x_3+x_4$. There are 6, and 6 divides 24. The swaps that do not change $x_1+x_2$ number $24\div6=4$: doing nothing, swapping $x_1$ with $x_2$, swapping $x_3$ with $x_4$, and doing both swaps.

Exercise: is there an expression in four letters that takes exactly 5 values?

No. By Lagrange’s theorem, the number of values must divide $4!=24$, whose divisors are $1, 2, 3, 4, 6, 8, 12$ and $24$. The number 5 is not on the list, so no expression in four letters takes 5 values, however ingenious it may be.

Sources: Joseph-Louis Lagrange, Réflexions sur la résolution algébrique des équations (1770–1771), in volume 3 of the Œuvres de Lagrange · biography of Lagrange, MacTutor · Israel Kleiner, A History of Abstract Algebra (2007).

All Lagrange topics · ← Four squares are enough · The derivative without infinitesimals and the mean value theorem →

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