The Equation That Couldn’t Be Solved

நான் இந்தப் பதிவையும் தமிழில் எழுத முயற்சி கூட செய்யப் போவதில்லை.

I wrote about Men of Mathematics recently. One of my favorite chapters in that book is about Evariste Galois.

Galois’s life was a tragedy, but simply wonderful. He died as a 20 year youth. He couldn’t have worked on math for more than 3-4 years. In that short period, he made an impact that is still being felt. The theories he developed are still fruitful.

Galois’s life is the perfect illustration of Murphy’s law. Whatever could go wrong, went wrong for him. A brilliant student, he got fascinated by mathematics and started ignoring other subjects. His genius was recognized and he was encouraged to try to join Ecole Polytechnique, the premier French “college”, where Cauchy and Fourier taught. He was rejected, apparently because he did much of the solving in his mind and skipped several steps in answering questions. He was encouraged to submit a paper outlining his revolutionary ideas, Fourier takes the paper home and dies. The paper was lost. He re-submits the paper, this time Cauchy takes the paper, Cauchy is impressed, but mysteriously, no further progress. He re-re-submits his paper, this time Poisson takes it, but he couldn’t understand the paper and rejects it. Then Galois gets into politics, goes to jail, comes back, fights a duel and dies.

I will be honest. Galois’s life was just fascinating. But I couldn’t understand Galois’s math from that book. All I could get was that he developed Group theory, with which he proved that quintic equations and equations of higher degree couldn’t be solved by a formula. How? It was not clear at all.

Once I went to college, I had access to some books that explained Group theory. I even had a class in M.Tech. about groups and fields. But it was all about the “How” and “What” about of Group theory, not about “Why”. In other words, I got lecture after lecture (or chapter after chapter) about what a group is, how to test for groups, what a field is, what a normal subgroup is, what a maximal subgroup is and so on. I had no clue about why I am learning about this. If I had found a chapter (or lecture) about how this is being used to proved a quintic equation couldn’t be solved by a formula, that would have been enough. The books/professor were all about the theory, that’s it. I had serious, probably unjustified, doubts about whether my professor even knew how to use group theory to solve non-textbook problems.

And I stumbled on this book a couple of weeks back – The Equation That Couldn’t Be Solved – by Mario Livio. Livio was fascinated by Galois and set out to explain Galois’s work and his impact. I knew some of it from my theoretical classes, but to me the takeaway is the connection between quintic equation and group theory.

Livio does a decent job of explaining how math progressed from quadratic/cubic/quartic equations to the struggles with quintic. He has a chapter on Abel, who first proved that quintic equations cannot have a formulaic answer. Then Galois’s life is covered in a chapter. And then he explains the basics of group theory, how symmetry can be described by groups, and several applications.

I have to admit – I still don’t understand the proof 100%. There is a jump from a step to another which has gaps for me. But I am happy! I am nearer to understanding a problem that I gave up years ago!

You can get the ebook here. Recommended for people who want to understand one of achievements of math…

Category: Math

One thought on “The Equation That Couldn’t Be Solved

  1. I thoroughly enjoyed and thanks that I could download the book and glanced it and surely will read it later fully. With a glance I enjoyed
    1.the Galois and construction of words oil, gas etc
    2.Palindrome numbers because I am very much into it I always share my own (found myself)palindrome dates to others (private sharing only) like recently my speedometer turned into 02220 on the exact date of 02/2/20 which I have clicked and sent.
    3. When I tried with a date may be two three years before the mirror image also same when it kept it vertically but could not thought if this word mirror symmetry.
    4. Admired this cube movement

    Liked by 1 person

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