Twenty-seven — A Brief Discussion on "Mathematical Girl 5"

"Mathematical Girl 5" is a "mathematics book disguised as a novel" that I came across while reading articles about elliptic curves over $\mathrm{GF}(p)$. Back in high school, ymq had mentioned Galois, the mathematician who died young. Coincidentally, while buying textbooks for the new semester, I spotted this book at a second-hand bookstore (the earlier volumes of the series were also there, but when I checked the next day, they were gone—quite a fateful encounter). The original price was 69, but with a discount, it came down to 15, so I bought it without hesitation.

In middle school, I had read Saruhiko Shimada's "Learning Mathematics, It's That Simple," which served as an introductory text to calculus and linear algebra, giving me an intuitive understanding of these two university-level courses. This book stood in stark contrast to the Tongji University versions of these textbooks (those who know, know), so I've always had a natural affinity for popular science mathematics books written by Japanese authors. "Mathematical Girl 5" is also written by a Japanese programmer, and it seems targeted mainly at engineering students (to some extent, it feels tailor-made for me). I figured that over the next few days, I would have a great reading experience.

And indeed, that night, I devoured half the book in one sitting. The earlier parts were quite basic, mainly introducing concepts (permutations, symmetric polynomials, groups, fields, conjugates, linear spaces). But the chapter proving the angle trisection problem really blew my mind (though the proof wasn't particularly elegant, it gave me a thrilling sense of satisfaction). The last time I felt this exhilarated was back in ninth grade when I solved the inverse of the half-angle problem.

The latter half of the book increased in difficulty (degree of field extensions, normal extensions, normal subgroups, splitting fields, cosets, normal decomposition). The book briefly re-proved the angle trisection problem using the concept of extension degrees and also accomplished something I had wanted to do since high school but never got around to—deriving the cubic formula. The part where Tetora claimed a "discovery," only for it to be debunked later, leading to her hurried departure, made me laugh—it felt so real!

My reading speed is inversely proportional to both time and the abstractness of the content. Initially, I thought I could finish the book in five days, but it took me over ten days to reach the final chapter on Galois theory. Then I realized that the previous nine chapters were essentially building up to this one. This chapter briefly covered Galois's first paper, judging the solvability of equations by reducing the Galois group and extending the coefficient field. Overall, it felt like an abstract version of Chapter 7, where the cubic formula was derived. Since I had forgotten some of the earlier content by this point, I had to constantly refer back, making the process somewhat painful. Thankfully, the author went easy on us and didn't throw the entire Galois theory at us, otherwise, I wouldn't have lived to write this article.

As for the romantic subplot in the novel—well, let's just say Miruka definitely had feelings for "me," but unfortunately, it was all in vain. "I" actually had a chance with both Miruka and Tetora, but the epilogue mentions that "I" became a math teacher, implying that neither worked out. Such an ending is rather bittersweet.

Finally, a word about the back cover:

It has a certain sophistication—you can't really understand its meaning without finishing the book. Only after completing it did I realize that this formula represents the necessary and sufficient conditions for judging the solvability of an equation using the Galois group, with a touch of love (or love triangle) symbolism. I have to admire the editor's thoughtful design.