Galois Groups and the Symmetries of Polynomials

Re:Galois Groups and the Symmetries of Polynomials by abetusk on Wednesday August 04, 2021 @09:26AM (#28060752)

My education on groups is by far and away deficient but one of the better books I came across is called "Fundamental Algorithms for Permutation Groups" by G. Butler [0]. It's kind of old (1991) but, in my opinion, excellent, especially for some one like me that's concerned with computation, run-time, algorithms, etc.

There's also a Akos Seress book, "Permutation Group Algorithms", which fills in some mathematical details and proofs but, notably, doesn't have one line of pseudo-code in the book.

Both focus on algorithms, and permutation groups in particular, which is just a small subset of group theory.

[0] https://www.springer.com/gp/book/9783540549550

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Re:Galois Groups and the Symmetries of Polynomials by sdenton4 on Wednesday August 04, 2021 @10:37AM (#28061519)

For the permutation group in particular, it's hard to beat Sagan's 'The Symmetric Group.' It's a bit more recent, and covers lots of really beautiful topics, and really shows off the nice connections between the combinatorics of S_n and the representation theory.

https://www.springer.com/gp/book/9780387950679

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Re:Galois Groups and the Symmetries of Polynomials by abetusk on Wednesday August 04, 2021 @03:59PM (#28065549)

Thanks for the recommendation.

Just briefly reviewing it, this is the kind of book I tend to shy away from. I'm sure it's a good reference for a particular type of group theory but I doubt I would use this type of book for anything but a reference.

I can't find any mention of "strong generating set" or algorithmic considerations of how to actually compute with permutation groups. As with many of the Springer books, it feels like a collection of scattered knowledge without a cohesive narrative.

It's completely possible I'm missing something with this book, especially since I haven't read it, and that for more mathematically minded folks, this book is a better introduction.

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