In order to explain what abstract algebra is, we have to go backwards in time a little. For most of human history, mathematics was viewed mainly as a tool to study science, and although some people may have focused more on the math side of things, the two ideas were inextricably joined: math was a tool used to explore science. The better they developed the tools, the better they could solve scientific problems. This was, for example, how calculus came to be; it was invented on a whim by Newton in order to solve physics problems, which were his prime area of concern. And it was this way across the disciplines, from geometry to analysis -- math was primarily done with the idea of accomplishing something tangible in mind.
But that all began to change in the mid-19th century, with the birth of an absolutely fascinating individual by the name of Evariste Galois. While in his teens, Galois became interested in math. He applied to Ecole Polytechnique, which was then and still is the MIT of France, and was rejected. He went to another French university, Ecole Normale. This is where Galois started to rock the house. When he first began the work that would soon make him famous, he attempted to get some papers published. Both were judged by Cauchy, another all-time famous French mathematician, and he turned them down. The story goes that Cauchy's issue was clerical, and truly believed in Galois' work; nonetheless, it was another kick in the groin for a young Evariste.
At this point, Galois' father commits suicide, shortly before young Evariste reapplies to Polytechnique. Naturally, he gets rejected. Keep in mind, at this point Galois is a genuine prodigy. Most attribute this to a personal offense made by Galois to the examiner; some blame it on the examiner, others on Galois. Either way, Galois returns to Ecole Normale, defeated, to continue his studies. But don't worry, much more bad luck lay in store for him there.
Galois essentially invented the field of abstract algebra, coined the mathematical term "group" (as in group theory) and solved the extremely long-standing problem of determining the necessary and sufficient conditions for a polynomial to be solvable by radicals. Basically, this means that (among other things) he figured out how to tell whether we could solve a particular polynomial equation with an elegant little formula like the "quadratic formula" from elementary school.
A bit more confident, he resubmit his work to the Academy, which Cauchy had rejected some years before. In fact, he submitted it directly to Fourier (of "Fourier transform" fame) but in line with Galois' previous luck, Fourier died soon thereafter and never reviewed it. That year, the prize was given to Jacobi and Abel, two highly deserving mathematicians. Nevertheless, Galois did succeed in getting two important papers published that gave birth to Galois theory. Another French mathematical all-star, Poisson, soon asked him to submit his work on Galois theory.
Now that something good had happened, Galois was apparently anxious to have more problems. So when the Trois Glorieuses came about, he became enthralled with the tumult of contemporary French politics, quit mathematics and joined a batallion of the National Guard. At a banquet, he gave a toast to the king that was perceived as a threat on his life and he was jailed the next day. By Bastille Day, he was released, but then promptly rearrested when he showed up to a protest covered in firearms. He was imprisoned, and shortly after arriving he received word from Poisson. Poisson derided his work as "incomprehensible" and said that his "argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor," which is about as a tough a statement as a mathematician could make. Galois, naturally, was of poor humor during his initial days of incarceration.
Tragically, his last words made a bleak reference to dying at the age of twenty. An obvious end to a man whose life was fraught with constant neverending disaster, which makes it all the more shocking how influential his work has become. And Galois' last word on his work? Bitterly, he says, "ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess."
What was in this work, and why was it so vastly important? Galois was trying to study a problem, which basically amounts to determining whether a polynomial equation (like 4x^2=0 or 5x-1=0 or x^10-x+1=0) admits a solution. It turns out the answer can be yes or sometimes. They can always be solved for degree less than five. But higher than four (for a polynomial like x^5+x-1=0) how can we tell, for one particular such equation, in some kind of mathematically sound way, whether it's solvable -- and how? The theory for this was laid by Galois in his short life's work. He did a few amazing things. Like Lagrange before him, he connected permutations and the roots of the polynomials (roots meaning "solutions"). What he realized was that he had to look at those permutations of the roots that had a remarkable property: any algebraic equation of the roots (with rational coefficients) satisfied before the permutation is also satisfied after. He then makes another incredible step, formalizing the notion of a "group" as we know it.
I've put off telling you what a group is but it's the first thing in abstract algebra. A group is a set (like {1,2,3} or {a,b,c} or "all the colors in the world") together with an "operation" such that taking any two elements, and doing the "operation" on them, gives you another element in the group. Pretty simple, but pretty powerful. Rich structure can be induced from these simple criteria, based on the rules of the operation and the elements you choose. The shocking part of abstract algebra is that given two completely unrelated groups, one can define homomorphisms (which you can think of as teleports between two different mathematical objects) to go between them. That means one can apply thinking and reasoning about one space to another.
Galois noticed that permutations formed such a group. And in fact, each polynomial has something called a "Galois group". The structure of the group corresponding to a polynomial (called that polynomial's Galois group) tells you whether or not a solution exists.
What does structure mean?
This is an idea that must resonate. It is so crucial, and such a powerful concept that is, I believe, a large organizing theme in the search for human knowledge. In this sense, structure simply means that a group has particular "subgroups" -- that is, groups within the group that are themselves also a group. So for example the even integers (under addition) make a subgroup of the integers (under addition). There are many interesting things that follow from this, and basically the subgroups are what define the structure of the group. And this is the case in Galois' solution, as well.
By beginning the trend towards abstraction and generalization, Evariste Galois set mathematics on a course to decouple from the physical sciences and become a science in its own right. This begs the question -- exactly what "science" do mathematicians study? I believe they study reality. They study some kind of general fabric of our universe so fundamental to existence that it perhaps goes unnoticed -- it is a concept too vague to give a simple label, like "chemistry" or "physics." I suppose that's why we call it mathematics.
This was a beautiful post. Very interesting story. Can I ask what introduced you to Galois?
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