The very mention of the name Galois to a mathematics undergraduate or recent graduate will either result in them wincing or being completely and utterly jealous. You see, before he reached the age of most mathematics graduates - about 21 years old - Monsieur Galois had constructed a branch of mathematics that solved a problem that had been around for centuries. And, true to form for a genius, he certainly had many downs in his short but ultimately productive life.
Life and Achievements - a Brief History
Evariste Galois was born on 25 October, 1811, the son of the mayor of the village of Bourg-la-Reine just outside Paris. He was taught at home by his mother until he reached the age of 12, and then he entered school. To start off with, he was a star pupil. However, he soon became bored with life at the school. He didn't enjoy his classical studies, and this lead to him having plenty of trouble with the school authorities, who described him using words such as 'eccentric' and 'argumentative'. His boredom was only relaxed when he found books he was interested in - books by famous mathematicians such as Legendre, Lagrange and Abel - and he understood and mastered such books before he was 16.
At the age of 16, he took the entrance exam for the prestigious Polytechnique in Paris, but failed miserably. Years later, after his death, it was remarked that:
A candidate with superior intelligence is lost with an examiner of inferior intelligence1.
However, he did find a teacher in Louis Richard, and a year later, he produced his first paper. The year after that, at the age of 18, Galois re-applied to the Polytechnique. Unfortunately, the exam went even worse this time round - Galois lost his patience with an examiner during the oral part of the exam and threw an eraser at him, scoring a direct hit. Of course, this meant that he could never re-apply again.
A year later, he attended the university instead, and he submitted the three original papers he had written himself on algebraic equation theory to the Academy of Sciences. What happened next is probably the most unfortunate episode of all. Cauchy, who at the time was Secretary of the Academy, took the papers home with him to read. A short time later, while he was in the middle of writing a report on them, he was suddenly taken ill and died. In the midst of all the confusion surrounding his death, the papers written by Galois and the report on them by Cauchy were lost, and have never been recovered2. Quite unsurprisingly, Galois was rather annoyed and upset at this further turn of events against him - he was quoted at the time as saying:
Genius is condemned by a malicious social organisation to an eternal denial of justice in favour of fawning mediocrity.
Strong words for such a young man.
'I have not time. I have not time'
And so we come to 1830. The French masses revolted, and Galois was a staunch supporter of their stance. So much so that he wrote a letter condemning the director of his school, and was consequently expelled. He tried to start his own school - this failed miserably too. Then he joined the National Guard, but this led to six months in jail for 'illegally wearing a uniform' - probably one of the more bizarre sentences of the day.
Thus we are brought to the final grand misadventure of the short life of Galois. When he was released from prison, the authorities and his political opponents were very aware of his hot-headed nature and were on the look-out for his next move. Then he met Stephanie du Motel, and promptly fell in love with her. However, as in his life previously, this was another bad move. Unfortunately one of his old political adversaries was also 'involved' with M'selle du Motel - leading to them both fighting for her affection. In fact, such a fight that on 30 May, 1832, they went out to duel with pistols at 25 paces. Galois had no real chance in such a duel, which ended with him fatally wounded and left to die by his adversary. A peasant found him hours after he was shot and took him to hospital, where he died the following day in the arms of his brother, having only lived 20 years and seven months.
However, in the nights directly before the duel, he had disclosed all the mathematics that was clear in his head to his friend Augustine Chevalier in a now-famous letter. Throughout this letter the poetic line 'I have not time. I have not time' is scrawled in Galois' characteristically messy handwriting. Galois stayed awake for as long as possible during these evenings, writing down everything that was 'clear in his head'. As well as the results that had been mysteriously lost around the time Cauchy died, he also wrote down some other mathematics that had been in his head for some time. He annotated and corrected some of his own papers that had previously been rejected - here is one such example. Poisson had left the following note in the margin of one of his papers:
The proof of this lemma is not sufficient. But it is true according to Lagrange's paper, No 100, Berlin 1775.
Directly beneath this, Galois wrote:
This proof is a textual transcription of that which we gave for this lemma in a memoir presented in 1830. We leave as an historic document the above note which M Poisson felt obliged to insert.
Bitter? Yes. And he had every reason to be as well. At one point during those few days he even had to write:
There are a few things left to be completed in this proof. I have not the time.
His letter to Chevalier ended with a plea to his friend of many years to get his papers sent on to an eminent mathematician of the time for their 'opinions not as to the truth but as to the importance of these theorems... I hope some men will find it profitable to sort out this mess.' Eventually, the papers fell into the hands of Liouville, and the rest, as they say, is history.
What is intriguing to many mathematicians is the following - what mathematics was unclear in the head of such a genius? Had he invented an even better way of observing the clever links between groups and field extensions?
What is clear nowadays, however, is that his insight helped future generations to a better understanding of the concept of a link between groups and field extensions. Chevalier kept everything that Galois had sent him, and eventually copies of the papers found their way to Joseph Liouville, an eminent French mathematician of the 19th Century. After a few months and very slight correction and reproof from Liouville, Galois' results were announced to the mathematical world by Liouville himself in 1843, and published by him in 1846.
A Brief Explanation of Some of his Most Important Mathematical Work
What Galois wrote in his last few living days is now known as The Fundamental Theorem of Galois Theory. The theorem provides a simple link between the basic mathematical concepts of groups and field extensions. It may sound like nothing important to your average everyday person, but to a mathematician, a link between something very abstract and not-at-all understood (in this case a field extension) and something abstract but well-studied and understood (in this case a group) is a very useful thing indeed.
The most important feature of this mathematical work by Galois was that it gave a solution to the problem of finding the roots of a quintic (or indeed higher order) equation - an equation where the highest power of the variable is 5 (or higher). Mathematicians already knew how to solve for the roots of a quadratic equation in terms of the constants that multiplied each term (see the formulae in Imaginary Numbers for this), and they also knew more complicated formulae to find the roots of a cubic equation of the form:
ax3 + bx2 + cx + d = 0
and a quartic equation of the form:
ax4 + bx3 + cx2 + dx + e = 0
in terms of the coefficients of the powers of x (in this case a, b, c, d and e). These were collectively known as solutions by radicals of such equations. A radical is a number which can be obtained solely from a combination of addition, subtraction, multiplication, division, and taking n-th roots, where n is a positive integer (ie 2, 3, 4, etc).
No such solution to find the roots in terms of these so-called radicals had been found in general for the quintic equation - an equation of the form:
ax5 + bx4 + cx3 + dx2 + ex + f = 0
In this case, certain special cases had been shown but that was all. What Galois theory proved was that there was no general solution to find the roots of a quintic equation in terms of radicals. Pretty impressive stuff for someone who died before reaching the age of 21.