Remember the quadratic formula from high school? This is the formula used to find the roots of a polynomial with degree 2 such as . Similar formulas exist to find the roots of polynomials of degree 3 and 4. But there is no analogous formula for degree 5 polynomials. Though this knowledge has a long and complex history, the result is largely understood due to efforts of one tragic man Évariste Galois.
In 1928, while Galois was seventeen years old, he failed the entrance examination to the École Polytechnique, the most prestigious institute of mathematics in France at the time. He instead attended The École Normale. While here, he began making fundamental discoveries related to the theory of polynomials and submitted two papers on the topic to the Academy of Sciences. The eminent mathematician Augustin Cauchy refereed these papers but did not accept them for publication. Whether Cauchy recognized the importance of these papers is disputed, however he did recommend that Galois submit his papers to another great mathematician Joseph Fourier. Galois did, though unfortunately, Fourier died shortly afterwards and the paper was lost.
Galois' time at the École Normale coincided with the overthrowing of King Charles X. During the ensuing revolutionary activity, the students of the École Normale were locked in by the school director. In reaction, Galois penned a scathing critique of the director, submitting it to the Gazette de Écoles. The letter resulted in his expulsion.
Following this, Galois became further involved politically. In fact, he joined an artillery, which was quickly disbanded for fear that they would destabilize the new government. Several months later, a heavily armed Galois led a Bastille Day protest while dressed in the uniform of this banned artillery. For this, he was arrested and spent roughly nine months in prison. During this time, he continued to develop his mathematical ideas and, at the suggestion of Poisson who rejected another of Galois' papers submitted prior to his arrest, collect and refine his manuscripts. Galois was released in April of 1832 and one month later, would be dead.
As many had before him, Galois fought in a duel. The reasons for this duel are not clear, though it has been speculated that it may have been the result of a quarrel over a girl he had fallen for. The evening prior to the duel, Galois stayed up all night composing a letter to friend and noted mathematician Auguste Chevalier. This letter expounded his mathematical ideas. The next morning, he was shot in the abdomen and, shortly thereafter, passed away. He was twenty years old.
The ideas contained in this letter were finally published thirteen years later. The most famous content was a proof that there is no general formula to solve for the roots of a polynomial of degree five. Though a flawed proof of this fact had been published decades earlier, the methods used by Galois were original and led to deep and interesting research in a sub-field of algebra now known as Galois Theory.