This is a question where historical context helps a lot, I think.<\/p>\n
Certainly, one can write about the specific results that Galois proved without any reference to the context in which they appeared, and they are beautiful. I really love Galois theory<\/p>\n
. But I fear that if one asks about the\u00a0applications\u00a0of Galois theory, the answers may seem very niche. And that does a grave injustice to a man (barely a man\u2014he was only 20 when he died) who helped revolutionize how we think about mathematics.<\/p>\n
So, let\u2019s set the scene a little.<\/p>\n
The 19th century was a time of revolution in Europe. The political revolutions of that time are certainly more famous, but it was a tumultuous time in science and mathematics as well.<\/p>\n
Prior to the 19th century, mathematics was mostly very concrete. Whatever mathematical objects were considered were primarily pulled from real-world experience: all of Euclidean geometry arose from physical drawings with straightedges and compasses; the real numbers arose from ideas about magnitudes and measurements;\u00a0e<\/span><\/span><\/span><\/span>e<\/span><\/span><\/span>\u00a0arose from studies of compound interest, and so on, and so on. What didn\u2019t come from such concrete considerations was viewed with suspicion\u2014complex numbers, for instance, were often treated as black magic, despite the fact that the rules for working with them are dead simple and they are incredibly useful.<\/p>\n In the 19th century, all of this changed. Mathematicians began exploring objects that didn\u2019t seem like they could be pulled from real-world experience, even if you could describe their mathematical properties perfectly well. This was the era in which non-Euclidean geometry was discovered<\/p>\n , for example. While, today, we recognize that hyperbolic geometry is, in some ways, even\u00a0more\u00a0closely related to the actual geometry of space than Euclidean geometry is<\/p>\n , at the time it seemed like something utterly devoid of application. Mathematicians as a whole had to be dragged kicking and screaming to embrace these kinds of ideas, but ultimately they did, and the field became so much richer for it.<\/p>\n And at the same time as Gauss, Lobachevsky, Bolyai, and Riemann were reinventing what it means to study geometry, Galois was doing the same thing for algebra.<\/p>\n Picture it. It\u2019s France, 1829. Politically, it is tense: a year from now, Charles X will be deposed\u2014to be replaced by his cousin, Louis Philippe I. But, for the moment, there is something more pressing. A young man\u2014only 17 years old\u2014storms out from his examinations; he has been denied entry to the prestigious \u00c9cole Polytechnique for a second time. He finds his examiners inane and plodding; they find him sloppy and explosively hot-headed. He is filled with fire and rage; his father had committed suicide just days before. His head is bubbling with ideas that are decades ahead of his time.<\/p>\n This is \u00c9variste Galois.<\/p>\n He had already published a paper the previous year; he would publish three more over the next few years while in and out of prison. (He was a fierce republican and made no attempts to hide his disdain for the monarchy.) In 1832, at just 20 years old, he was pulled into a duel and shot to death.<\/p>\n Having now a sense of the man and the environment in which he was, let\u2019s actually discuss his work.<\/p>\n What was algebra prior to the work of Galois? As a field, it was almost entirely about solving equations. When can one get write down a solution to a polynomial equation in terms of radicals? When can one get integer solutions to such an equation? These were the kinds of questions that mathematicians were asking (and sometimes answering). To the extent that new algebraic objects were introduced, it was solely in the service of such concrete goals\u2014this is how complex numbers came to be, and modular arithmetic, and so on.<\/p>\n Galois did produce results and solutions in this vein, but he also intuitively understood\u2014in a way almost no one else did at that time\u2014that one could study these algebraic objects themselves, their properties, and how they related to one another. When he published his paper on what is now called Galois theory, it did provide a way to prove that the roots of various polynomials cannot be written down in terms of radicals, yes\u2014but Galois himself wrote that this was merely an\u00a0application\u00a0of the theory, and not what it was principally about. This was completely missed by his contemporaries, who tried to make sense of Galois\u2019 work by focusing on this one element that they judged as actually important. It was only decades later that this fundamental shift in perspective was understood and appreciated.<\/p>\n The core of Galois theory\u2014what he is principally remembered for\u2014can be roughly explained thus, in modern language:<\/p>\n<\/blockquote>\n There are algebraic structures called groups and fields, and there is a fundamental connection between the two that allows you to transform difficult problems about one into easy problems about the other.<\/p>\n Of course, this is a little anachronistic, since Galois did not have a modern definition of either groups or fields\u2014those didn\u2019t come until the 1880s and 1890s, respectively. He still thought about them in much more concrete terms. But if I may be allowed to continue this anachronism for the ease of exposition, I can give a little insight into what these are.<\/p>\n What are\u00a0groups? These are pretty much the most fundamental algebraic structures there are<\/p>\n![](\"https:\/\/qph.cf2.quoracdn.net\/main-qimg-378d5e895b4824111dba1119c2bc76cd.webp\")
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