Today is the 200th birthday of Evariste Galois, who did not live to celebrate his 21st, but found time in his short 20 years to develop a circle of ideas that permeate modern mathematics. We know of these ideas because Galois spent the night of May 30, 1832 scribbling them furiously in a letter to a friend, in advance of the fatal duel he would fight the following morning. According to the great mathematician Hermann Weyl, “This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.”
(If this were a less serious post, I might suggest that this famous letter was the first example of a Galois Correspondence.)
Now, two centuries later, every first year graduate student in mathematics spends a semester studying Galois Theory, and many devote their subsequent careers to its extensions and applications. Many of the greatest achievements of modern mathematics (for example, the solution to Fermat’s Last Theorem) are, at their core, elucidations of Galois’s 200-year-old insight.
As every high school student knows (or should know), a quadratic equation (like, say, x2 – 4x – 1 = 0) can be solved by applying the quadratic formula (which, in this case, gives x = 2 ± √5). The quadratic formula uses only addition, subtraction, multiplication, division, and the extraction of square roots.
What about a cubic equation, like, say, x3 + 2 x2 – 5 x – 3 = 0 ? The less well-known cubic formula finds the solutions, using only addition, subtraction, multiplication, division, and the extraction of square and cube roots.
And, yes, there’s a quartic formula, for equations of degree 4. But it stops there. Galois’s contemporary Niels Abel (who, unlike Galois, survived to the ripe old age of 26) showed that no formula can consistently solve equations of degree 5 using only addition, subtraction, multiplication, division, and the extraction of roots.
On the other hand, some equations of degree 5 and higher can be solved by such formulas. Call those equations solvable. Galois figured out how to identify the solvable equations. It all comes down to understanding symmetry. Galois was the first to see clearly that the solutions to any equation satisfy certain symmetries. (For example, the solutions 2-√5 and 2+√5 are symmetric under the interchange of the plus and minus signs on the square root.) The nature of those symmetries differs from equation to equation, and dictates whether the equation is solvable. This in turn leads to a much deeper appreciation of the importance of symmetries throughout the theory of equations and throughout algebra more generally.
Today, algebraists take it for granted that understanding an equation, or a system of equations, entails understanding its symmetries. The development of that instinct was a key advance in the history of thought. After almost two centuries, we still use it to discover new insights and to solve old problems every single day.