Fermat's Last Theorem (33 page)

My Dear Friend,

I have made some new discoveries in analysis. The first concern the theory of quintic equations, and others integral functions.

In the theory of equations I have researched the conditions for the solvability of equations by radicals; this has given me the occasion to deepen this theory and describe all the transformations possible on an equation even though it is not solvable by radicals. All this will be found here in three memoirs …

In my life I have often dared to advance propositions about which I was not sure. But all I have written down here has been clear in my head for over a year, and it would not be in my interest to leave myself open to the suspicion that I announce theorems of which I do not have a complete proof.

Make a public request of Jacobi or Gauss to give their opinions, not as to the truth, but as to the importance of these theorems. After that, I hope some men will find it profitable to sort out this mess.

I embrace you with effusion,

E. Galois

The following morning, Wednesday 30 May 1832, in an isolated field Galois and d'Herbinville faced each other at twenty-five paces armed with pistols. D'Herbinville was accompanied by seconds; Galois stood alone. He had told nobody of his plight: a messenger he had sent to his brother Alfred would not deliver the news of the duel until it was over and the letters he had written the previous night would not reach his friends for several days.

The pistols were raised and fired. D'Herbinville still stood, Galois was hit in the stomach. He lay helpless on the ground. There was no surgeon to hand and the victor calmly walked away leaving his wounded opponent to die. Some hours later Alfred arrived on the scene and carried his brother to Cochin hospital. It was too late, peritonitis had set in, and the following day Galois died.

His funeral was almost as farcical as his father's. The police believed that it would be the focus of a political rally and arrested thirty comrades the previous night. Nonetheless two thousand republicans gathered for the service and inevitably scuffles broke out between Galois's colleagues and the government officials who had arrived to monitor events.

The mourners were angry because of a growing belief that d'Herbinville was not a cuckolded fiancé but rather a government agent, and that Stéphanie was not just a lover but a scheming
seductress. Events such as the shot which was fired at Galois while he was in Sainte-Pélagie prison already hinted at a conspiracy to assassinate the young trouble-maker, and therefore his friends concluded that he had been duped into a romance which was part of a political plot contrived to kill him. Historians have argued about whether the duel was the result of a tragic love affair or politically motivated, but either way one of the world's greatest mathematicians was killed at the age of twenty, having studied mathematics for only five years.

Before distributing Galois's papers his brother and Auguste Chevalier rewrote them in order to clarify and expand the explanations. Galois's habit of explaining his ideas hastily and inadequately was no doubt exacerbated by the fact that he had only a single night to outline years of research. Although they dutifully sent copies of the manuscript to Carl Gauss, Carl Jacobi and others, there was no acknowledgment of Galois's work for over a decade, until a copy reached Joseph Liouville in 1846. Liouville recognised the spark of genius in the calculation and spent months trying to interpret its meaning. Eventually he edited the papers and published them in his prestigious
Journal de Mathématiques pures et appliquées.
The response from other mathematicians was immediate and impressive because Galois had indeed formulated a complete understanding of how one could go about finding solutions to quintic equations. First Galois had classified all quintics into two types: those that were soluble and those that were not. Then, for those that were soluble, he devised a recipe for finding the solutions to the equations. Moreover, Galois examined equations of higher order than the quintic, those containing
x
⁶
,
x
⁷
, and so on, and could identify which of these were soluble. It was one of the masterpieces of nineteenth-century mathematics created by one of its most tragic heroes.

In his introduction to the paper Liouville reflected on why the young mathematician had been rejected by his seniors and how his own efforts had resurrected Galois:

An exaggerated desire for conciseness was the cause of this defect which one should strive above all else to avoid when treating the abstract and mysterious matters of pure Algebra. Clarity is, indeed, all the more necessary when one essays to lead the reader farther from the beaten path and into wilder territory. As Descartes said, ‘When transcendental questions are under discussion be transcendentally clear.' Too often Galois neglected this precept; and we can understand how illustrious mathematicians may have judged it proper to try, by the harshness of their sage advice, to turn a beginner, full of genius but inexperienced, back on the right road. The author they censured was before them ardent, active; he could profit by their advice.

But now everything is changed. Galois is no more! Let us not indulge in useless criticisms; let us leave the defects there and look at the merits …

My zeal was well rewarded, and I experienced an intense pleasure at the moment when, having filled in some slight gaps, I saw the complete correctness of the method by which Galois proves, in particular, this beautiful theorem.

At the heart of Galois's calculations was a concept known as
group theory
, an idea which he had developed into a powerful tool capable of cracking previously insoluble problems. Mathematically, a group is a set of elements which can be combined together using some operation, such as addition or multiplication, and which satisfy certain conditions. An important defining property of a group
is that, when any two of its elements are combined using the operation, the result is another element in the group. The group is said to be
closed
under that operation.

For example, positive and negative whole numbers form a group under the operation of ‘addition'. Combining one whole number with another under the operation of addition leads to a third whole number, e.g.

Mathematicians state that ‘positive and negative whole numbers are closed under addition and form a group'. On the other hand the whole numbers do
not
form a group under the operation of ‘division', because dividing one whole number by another does not necessarily lead to another whole number, e.g.

The fraction
¹
⁄
₃
is not a whole number and is outside the original group. However, by considering a larger group which does include fractions, the so-called rational numbers, closure can be re-established: ‘the rational numbers are closed under division'. Having said this, one stills needs to be careful because division by the element zero results in infinity, which leads to various mathematical nightmares. For this reason it is more accurate to state that ‘the rational numbers (excluding zero) are closed under division'. In many ways closure is similar to the concept of completeness described in earlier chapters.

The whole numbers and the fractions form infinitely large groups, and one might assume that, the larger the group, the more interesting the mathematics it will generate. However, Galois had a ‘less is more' philosophy, and showed that small carefully constructed groups could exhibit their own special richness.
Instead of using the infinite groups, Galois began with a particular equation and constructed his group from the handful of solutions to that equation. It was groups formed from the solutions to quintic equations which allowed Galois to derive his results about these equations. A century and a half later Wiles would use Galois's work as the foundation for his proof of the Taniyama–Shimura conjecture.

To prove the Taniyama–Shimura conjecture, mathematicians had to show that every one of the infinite number of elliptic equations could be paired with a modular form. Originally they had attempted to show that the whole DNA for one elliptic equation (the
E
-series) could be matched with the whole DNA for one modular form (the
M
-series), and then they would move on to the next elliptic equation. Although this is a perfectly sensible approach, nobody had found a way to repeat this process over and over again for the infinite number of elliptic equations and modular forms.

Wiles tackled the problem in a radically different way. Instead of trying to match all elements of one
E
-series and
M
-series and then moving on to the next
E
-series and
M
-series, he tried to match one element of all
E
-series and
M
-series and then move on to the next element. In other words each
E
-series has an infinite list of elements, individual genes which make up the DNA, and Wiles wanted to show that the first gene in every
E
-series could be matched with the first gene in every
M
-series. He would then go on to show that the second gene in every
E
-series could be matched with the second gene in every
M
-series, and so on.

In the traditional approach one had an infinite problem, which was that even if you could prove that all of one
E
-series matched all of one
M
-series, there were still infinitely many other
E
-series and
M
-series to be matched. Wiles's approach still involved tackling infinity because even if he could prove that the first gene of
every
E
-series was identical to the first gene of every
M
-series there were still infinitely many other genes to be matched. However, Wiles's approach had one major advantage over the traditional approach.

In the old method, once you had proved that the whole of one
E
-series matched the whole of one
M
-series, you then had to ask, Which
E
-series and
M
-series do I try and match up next? The infinity of
E
-series and
M
-series have no natural order and so whichever one is tackled next is a largely arbitrary choice. Crucially, in Wiles's method, the genes in the
E
-series do have a natural order, and so having proved that all the first genes match (
E
₁
= M
₁
), the next step is obviously to prove that all the second genes match (
E
₂
=
M
₂
), and so on.

This natural order is exactly what Wiles needed in order to develop an inductive proof. Initially Wiles would have to show that the first element of every
E
-series could be paired with the first element of every
M
-series. Then he would have to show that if the first elements could be paired then so could the second elements, and if the second elements could be paired then so could the third elements, and so on. He had to topple the first domino, and then he had to prove that any falling domino would also topple the next one.

The first step was achieved when Wiles realised the power of Galois's groups. A handful of solutions from every elliptic equation could be used to form a group. After months of analysis Wiles proved that the group led to one undeniable conclusion – the first element in every
E
-series could indeed be paired with the first one in an
M
-series. Thanks to Galois, Wiles had been able to topple the first domino. The next step of his inductive proof required him to find a way of showing that if any one element of the
E
-series matched the corresponding element in the
M
-series, then so must the next element match.

Getting this far had already taken two years, and there was no hint of how long it would take to find a way of extending the proof. Wiles was well aware of the task ahead: ‘You might ask how could I devote an unlimited amount of time to a problem that might simply not be soluble. The answer is that I just loved working on this problem and I was obsessed. I enjoyed pitting my wits against it. Furthermore, I always knew that the mathematics I was thinking about, even if it wasn't strong enough to prove Taniyama–Shimura, and hence Fermat, would prove something. I wasn't going up a back alley, it was certainly good mathematics and that was true all along. There was certainly a possibility that I would never get to Fermat, but there was no question that I was simply wasting my time.'