An 18 year old brilliant kid, who solved a 350 year old Mathematical problem gracefully and doing so, introduced an entirely new branch of Mathematics that redefined algebra, and changed how we look at nature - how do you think Life treated him?

Fame? Popularity? Awards? An extraordinary life?

Think again.

• Two failures in admission in his favorite institution.
• Three rejections by Mathematicians.
• A father who commited suicide.
• Separation from his family.
• Expulsion from school.
• Failed attempt at teaching.
• A failed love.
• A political turmoil that led to his imprisonment.
• Mental depression and lunacity.
• A duel which left him fatally wounded only to die at 20.

Meet Évariste Galois  , one of the brightest mind in Maths.

Before knowing what he achieved, let’s look at what the problem was.

Can you tell me the solution of $x+5=3$ ?

Easy right? Just subtract 5 from both sides and you get x=-2

Let’s move on a little higher.

What’s the solution of $x^2–5x+6=0$ ?

A little difficult but easy when you apply Sridhar Acharya’s formula  . The answer turns out to be $x=2$ and $3$.

See that in the first equation, the highest power of $x$ was $1$. And we had one solution.

In the second equation, the highest power of $x$ was $2$ and we had two solutions.

This highest power of $x$ is called the degree of the (polynomial) equation. The number of solutions of a (polynomial) equation is less than or equal to its degree. (exactly equal if you allow Complex numbers).

Fine?

Let’s move on.

What’s the solution of $x^3–2x^2+3x-5=0$?

This is a little difficult and out of high school Maths. We have several methods - the most famous being Cardano/Tartgalia method  , which is complicated but doable.

Let’s increase the degree a little.

$x^4–5x^3+x^2-x+2=0$

This one is way more complicated, and there are still methods, the most famous one is Ferrari’s method  , but we can still do it.

What about degree $5$?

$x^5-x^4+2x^3–8x^2+5x+9=0$

And that’s where Mathematicians got stuck, for about $350$ years. All the known methods of solving polynomial equations failed horribly.

So, how do we solve it?

Niels Henrik Abel, (who also had a tragic life) proved in 1829, that there is no method involving algebraic operators (addition, subtraction, multiplication, division and exponentiation) which can act as a general method for solving such degree $5$ equations, which we call a quintic.

But wait, clearly we can solve $x^5 = 32$ , right? The answer is trivially $x = 2$ (with some complex cousins, 4 to be precise).

So, which equations can be solved and which cannot be? Galois worked on this problem.

Évariste Galois was born on October 25, 1811, in Paris. His father Nicolas-Gabriel Galois was managing at the time a quite reputable school for boys in Bourg-la-Reine.

It was Napoleonic era and his family was loyal to the emperor. Galois’ uncle was even an Imperial Guard. During Napoleon’s brief return in power in 1815, Nicolas was appointed as the Mayor of Bourg-la-Reine, the position he held even after Waterloo.

Galois was very close with his father. They’d make witty poems together and spend time. A major part of Galois’ early education comes from his father.

October 1823, Galois left home to be admitted to Parisian boarding school Lycee Louis-le-Grand. The school was like a prison, and thanks to the ongoing revolution, riots and quarrel between students was pretty normal. Their day started at 5:30 and ended at 8:30 offered a little time for students’ enjoyment. A millitary level discipline was imparted on the students. Break even a slightest rule, and you’d be confined in a cell as a punishment.

When Galois came to the school, conservative Nicolas Berthot was appointed as school principal. The students took this as a step to impart Church’s control and refused to participate in the morning prayer and other customary actions. Overall a fire was burning inside the students.

For the first two years, Galois proved to be a success, but the environment took a toll on Galois. During 1825–26, Galois suffered a painful earache, which only added to the depressing state of his mind. The separation from his father, with whom he used to enjoy the exchange of witty couplets, was particularly hard on the young boy. Consequently, his schoolwork began to deteriorate.

In 1826, the newly appointed Headmaster Pierre-Laurent Laborie decided that Galois was too young for the rhetoric class which, according to him, required "judgment that only comes with maturity." In January, therefore, Galois was forced to repeat the third-year classes. The unpleasant experience with rhetoric, however, turned out to be a blessing in disguise—Galois discovered mathematics, thanks to his newly appointed teacher, Mr. Hippolyte Vernier.

By 1827, Galois had lost interest for all other subjects and devoted all his time on Maths. He threw away the conventional textbooks and went for research papers. A mathematician was born.

Unknowing to Abel’s work, he began working on quintic, and just like Abel, he also thought that he had discovered a formula, only to realise that he had made a mistake, and continued working.

However, as Vernier had predicted, in spite of his extra ordinary talent, he never studied systematically and methodically. He had extreme knowledge in some fields and almost zero in others. Not listening to Vernier’s advice, he tried to take the entrance exam to the legendary Ècole polytechnique  a year early, in 1828.

He failed. He lacked systematic preparation. His first major setback.

Forced to continue at Lycee Louis-le-Grand, Galois enrolled in the Special Mathematics class of Louis-Paul-Emile Richard, who recognised his brilliance, and engaged him in original research.

In 1829, the 17 year old Galois discovered what later proved to be the turning point of Algebra - Group Theory  . Let’s take a small detour and see briefly what this is all about.

Groups, are kind of like an algebraic structure. Without going into much Mathematical jargon, think of it like a set of some “things” and an “operation”. You can perform this “operation” on those “things”. If they hold some properties, they’d form a group.

Why is this important?

Groups allow us to redefine Algebra. Earlier, we could only operate on numbers, but groups made way for Abstract Algebra, which lets us do “stuff” on any “thing”.

What did Galois do?

He introduced the concept of Normal Subgroups, and to every equation, he attached a special group, called Galois group  . Think of it like a DNA code of that equation. Combined with the idea of symmetry, this Galois group lets us analyze an equation quickly.

Mind blowing, isn’t it? But it was too advanced for the world. The mathematicians didn’t expect a 17 year old to create an entire new branch of Maths to solve an algebra problem.

Richard encouraged Galois to put his theory in two memoires and published them in front of Cauchy  , one of the prominent Mathematician of all times, in May 25, and June 1 of 1829. It was entrusted for judgment to Cauchy, Joseph Fourier  , and mathematical physicists Claude Navier and Denis Poisson.

Galois’ life was about to take a turn, and it did, in unexpected ways.

More than six months after the submission, on January 18, 1830, Cauchy wrote the following apologetic letter to the academy -

I was supposed to present today to the Academy first a report on the work of young Galois, and second a memoir on the analytic determination of primitive roots in which I show how one can reduce this determination to the solution of numerical equations of which all roots are positive integers. Am indisposed at home. I regret not to be able to attend today's session, and I would like you to schedule me for the following session for the two indicated subjects.

The next session took place on January 25. But Cauchy was going through an egotistical phase. He presented only his memoire and never mentioned Galois’ work.

In June of 1829, the Academy of Sciences announced the establishment of a new Grand Prix for Mathematics. Tired of waiting for Cauchy's verdict, and having learned from Galois decided to resubmit the work, with some modifications, as an entry for the prize. The work was entered in February 1830, shortly before the March 1 deadline. The prize committee consisted of mathematicians Legendre, Poisson, Lacroix, and Poinsot. For reasons that are not entirely clear, the academy's secretary, Fourier, took the manuscript home. He died on May 16, and the manuscript was never recovered among his papers. Consequently, entirely unbeknownst to Galois, his entry was never even considered for the prize. The prize was offered to Abel posthumously (who had passed away in age 26 by the way) and Jacobi  .

Galois was furious to learn that his manuscript had been lost.

In July 2nd 1829, owing to the political turmoil, Galois’ father, who was accused of a scandal, thanks to a forgery by the new priest, commited suicide by gas ashpyxiation. Galois was emotionally devastated and broke down.

As fate had it, the second entrance to Ecole Polytechnique took place just one month after this mishap, on August 3. Galois was still mourning his father’s death.

Compared to Galois, the two examiners, Charles Louis Dinet and Lefebure de Fourcy, were, in historian E. T. Bell's words, "not worthy to sharpen his pencils."

He failed again.

Some say, due to his tendency of doing calculations in his head extremely fast and writing only the final result on the board caused his failure, but we don’t know for sure.

This failure left Galois devastated. He was forced to enter less prestigious Ecole Normale in Science Section.

In 1830, two of his articles were published in Ferrusac’s Bulletin. During this year, he met Auguste Chevalier, who later became his best friend.

In Ecole Normale, however, life was not easy for Galois. Thanks to the unstable times of revolution, the environment of education was absent. The students were forcefully barred from participating in the rebellion, by threatening to callin the troops.

Galois however wanted to be a part of the revolution. He turned into a revolutionary, ready to sacrifice himself for the revolution. After a letter against the director of Ecole Normale was published in a newspaper, although it was anonymous, Galois was suspected and expelled.

On January 2, 1831, the Gazette des ecoles published an article by Galois entitled "On the Teaching of the Sciences, the Professors, the Works, the Examiners." Most of Galois's complaints would sound relevant even today:

Until when will the poor youngsters be obliged to listen or to repeat all day long? When will they be given some time to reflect on this accumulation of knowledge, to be able to coordinate [find a pattern in] this endless multitude of propositions, in these unrelated calculations? . . . Students are less interested in learning than in passing their exams.

Galois lamented:

Why don't the examiners pose questions to candidates other than in a twisted manner? It seems that they fear being understood by those they are interrogating; what is the origin of this deplorable habit of complicating the questions with artificial difficulties?

With no school to study in, Galois had to support his family. He decided to give tuitions on Maths. But the course he chose was way advance for his students and his political activities took his attention, and his class was a failure.

At the beginning of 1831, Galois was asked to resubmit his paper to the academy. The new version was submitted on January 17, this time to be reviewed by Poisson and Lacroix.

More than two months had passed, with no words from the academy. Galois was extremely frustrated. He sent a letter to the president on March 31, 1831, adding sarcastically, "Sir, I would be grateful if you could relieve my concerns by inviting Mr. Lacroix and Mr. Poisson to announce whether they have also lost my memoir [as did Fourier], or whether they intend to report on it to the Academy.” but nothing helped.

The political turmoil and the events took a toll on Galois’ character. He became furious, kind of a lunatic. He lost his sense of the surroundings, and apparently took out a dagger, saying “This is how I will be sworn in to Louis-Philippe” in a banquet, which was percieved as a threat to the king and he was arrested.

Poisson and Lacroix finally presented their verdict on July 4, 1831 where they disapproved his propositions -

We have made all possible efforts to understand M. Galois's proof [of the conditions under which an equation is solvable by a formula]. His reasonings are neither sufficiently clear, nor sufficiently developed for us to be able to judge their exactness, and we are not in a position that enables us to give an opinion in this report. The author states that the proposition that makes the special topic of his memoir is a part of a general theory that could lead to many other applications. Frequently, it happens that different parts of a theory clarify one another, and are easier to grasp collectively, rather than when taken in isolation. One should therefore wait for the author to publish his work in its entirety to form a definitive opinion; but in the state at which the part submitted to the Academy currently is, we cannot recommend to you to give it your approval.

Galois was enraged at this decision. It was a huge blow to him. But this was not the end. He got engaged in Bastille day revolution and was sentenced to six months imprisonment. He was sent to Saint-Pelagie, prison for hardcores.

Most of the accounts of his prison life comes from his fellow inmates, where he is seen to mourn his father, agonized, bruised, and without any hope.

Galois's sister, Nathalie-Theodore, depicts the most heart-breaking picture of her brother's physical and mental state. After one distressing visit she writes in anguish in her diary:

"To endure five more months without a breath of fresh air! This is a very bad perspective, and I fear that his health will suffer much. He is already so tired. He does not allow himself to be distracted by any thought, he has taken a somber character that makes him age before his time. His eyes are hollow as if he is fifty years old."

In the spring of 1832, a cholera spread through Paris, and Galois was on March 13 from Saint-Pelagie to a convalescent home at 84-86, rue de Lourcine.

There he fell in love, with Stephanie Potterin du Motel, who lived in the same building of the convalescent home. Her father, Jean-Paul Louis Auguste Potterin du Motel, was a former officer in the Napoleonic army.

But the consequences were tragic.

Stephanie rejected his affections. Galois was not lovable. Life had turned him into a rough and rude creature. Things took a bad turn and She even denied having mere friendship with him.

Galois was devastated again. He wrote to Auguste -

My dear friend,

There is pleasure in being sad, if one can hope for consolation. One is happy to suffer if one has friends. Your letter full of apostolic grace has given me a little calm. But how can I remove the trace of such violent emotions as those which I have experienced? How can I console myself when I have exhausted in one month the greatest source of happiness a man can have? When I have exhausted it without happiness, without hope, when I am certain I have drained it for life?

He finishes by saying, “I’ll see you on June 1.”

He never saw Auguste again.

His death is surrounded by mystery. All we know that he was forced to perform a duel. He was released on May 25, and he faced his opponents on May 30.

We don’t know for certain who the opponents were. There had been many speculations, but it’s covered in mystery. But we know that Galois knew it was going to be his last day. He had lost all his will to live. He wrote goodbye letters to his friends and family.

He wrote to Auguste to collect his manuscript and ask Mathematician Jacobi to review them.

The duel happened on May 30, 1832. The exact events are not known, but Galois was shot in the stomach on the right. He was left to die before someone took him to the hospital.

His brother Alfred rushed to the hospital. In his last hours, Galois told a crying Alfred, “Don't cry, I need all my courage to die at twenty.”

He passed away at 10 am on May 31, 1832.

It took the world a long time to understand him. Thanks to Jacobi, Alfred and Auguste, his brilliance came to light.

His theory, apart from redefining algebra, coupled symmetry with algebra, which later proved to be the key to define geometry by symmetry (thanks to Klein). Basically, wherever there is symmetry, there is application of Galois theory.

His life, however, was one of the most tragic lives of all time. Had he been alive, he might have enriched Maths even more.

Life isn’t fair. Is it?

Credits to: “The equation that couldn’t be solved, by Mario Livio.”

As a note: Abel also had a tragic life, full of poverty. And coincidentally, he was also rejected by Cauchy and saved by Jacobi.