Évariste Galois was a prolific mathematician at a young age: at 17 he proved that no equation can exist which would solve 5th degree polynomials (the “quintic formula”). By 18, he was expelled from school: a rising leader in France’s 1830 July Revolution, Galois wrote a letter to his headmaster condemning the institution’s ban on students’ participation in the movement – and signed it, confidently, with his full name. By 19, he was in jail: at a banquet attended by the entirety of France’s political elite, Galois offered an ardent toast to the king – while, in a thinly veiled threat, holding a dagger above his cup! In prison, Galois submitted his groundbreaking mathematical work (on an early form of modern “group theory”) to France’s preeminent journal – only for it to be rejected as “incomprehensible”. By 20, Galois was dead, from wounds incurred in a duel fought regarding a mysterious “Stéphanie-Félicie Poterin”. Throughout the entirety of the night before the duel, anticipating his demise and working by candle-light, Galois compiled his final paper; eminent mathematician Hermann Weyl would one day argue that the “novelty and profundity” of Galois’s final work make it “perhaps the most substantial piece of writing in the whole literature of mankind.”
Group theory, and its containing field “abstract algebra”, are fascinating topics. Elementary school has taught us that there exist basic “operations” performed on numbers – addition, for example, or multiplication – that satisfy basic properties. These operations are both
- commutative: a+b = b+a, a×b = b×a, for any a, b, and
- associative: (a+b) + c = a + (b+c), (a×b)c = a(b×c) for any a, b, c.
Like much of mathematics, the study of groups seeks to generalize: the field investigates the properties shared by associative operations in general. Discarding assumptions of any kind about what exactly an operation does, group theory simply assumes that there exists an operation – an associative one, to be exact – defined on a set, which takes two elements in the set and “spits out” a third one. It also requires the existence of a neutral (identity) element and an “inverse” operation.
The consequences of these axioms, as simple as they are, are immense. Galois’ discovery relied, in part, on the observation that the operation of re-ordering (or permuting) a set of objects is, itself, an associative operation – and shares, by consequence, many of the properties held by numbers themselves. Studying the permutations of the roots of a polynomial equation, Galois linked the solvability of this equation to special properties in the group of permutations of its roots. (Whew!) His methods have been cemented and generalized into modern “Galois theory”, a subfield within abstract algebra.
Dying in the hospital the night of his duel, Galois’ last words to his younger brother Alfred were, “Don’t cry, Alfred! I need all my courage to die at twenty.” At 21, I’ve already outlived Galois. I’m neither an immortalized mathematical success nor an emblazoned leader of revolutionary street-protests. But hope is not lost. I travel to Russia this fall for a semester of study with the intensive “Math in Moscow” program at one of the premier mathematical centers in Russia (and the world). Some day, perhaps, I’ll rise to a level near that of Galois; some day, I hope, I’ll study graduate mathematics and make my own impact. If not, a legacy as a political hero would suffice. Long live the memory of Évariste Galois!