There's a story about Laplace, the great French mathematician. Much of his work was focused on "Celestial Mechanics", which is the mathematics of planetary orbits. At the end of his life, he decided to write a book summarizing all his discoveries. Unfortunately, he found that there were some results that he had proved many years ago, but now he had forgotten the proof, and couldn't reconstruct it. Whenever this happened, he dealt with the problem by writing, "It is easy to see that . . ."
For years my favorite piece of data (taped to my office door) had those words scribbled on the bottom*. So I was never the smartest in my class, not even close. But I do wonder if (worry that) we don't support, challenge, encourage, our best and brightest kids enough. Or maybe that is best left to the family? (I was going to write that bright kids are a precious resource, but that just sounds insulting to everyone.)
*Far-IR spectroscopy of semiconductors, a symmetry in the electron and hole wavefunctions allowed us to designate the exciton absorptions we were observing... details not important.
Galois theory seems to be one of those subjects that lends itself to attractive exposition. I had a similar experience when I was the same age with Joseph Rotman's book. And, I, too, saw much of myself and my own failings in Galois' mercurial, intense persona.
In theory the book has an appendix containing all the necessary prerequisites, so somebody with good mathematical maturity would be able to just take it off the shelf and get going.
In practice, I'd recommend at least some familiarity with abstract algebra, especially field theory.
Artin's book on algebra is well-regarded, and available for free online (http://home.ustc.edu.cn/~liweiyu/documents/Algebra,%20Second%20Edition,%20Michael%20Artin.pdf). A lot of people love this book for its heavy emphasis on matrix groups and linear transformations, but I sort of hate it for the same reason (and in particular, he takes a few hundred pages to get to the material that's of relevance to Galois theory).
Dummit and Foote (https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/0471433349) take more or less the opposite approach from Artin -- their work is quite abstract and dry, but encyclopedic, and introduces concepts in what I consider to be a sensible order. You should definitely not feel like you need to read all of this book to start on Cox -- the first few chapters, and then the material on field theory, should be more than sufficient.
If both of those options are too hard, the best really gentle introduction to abstract algebra that I know of is at the end of the second part of Harris and Hirst's book on discrete mathematics (https://www.amazon.com/Combinatorics-Graph-Theory-Undergraduate-Mathematics/dp/0387797106). They motivate the whole idea of group theory very nicely as part of a section on Polya's theory of counting, and the development is easy and natural.
For conceptual and philosophical material to hype you up on this journey, check out Hermann Weyl's "Symmetry" (https://www.amazon.com/Symmetry-Princeton-Science-Library-Hermann/dp/0691173257). Weyl is one of the 20th century's greatest mathematical physicists, and this is his book for laymen. He starts by analyzing different sorts of symmetry in nature and art, before gradually building up to the fully general abstract mathematical concept of symmetry that motivates group theory.
Thank you. I have very little real mathematical training or mathematical maturity. I read Euclid and Lobachevsky (and some Hilbert) at St. John's, but that's about all I have. I've been working through this book to start orienting myself: https://www.amazon.com/gp/product/0201102382
There's a story about Laplace, the great French mathematician. Much of his work was focused on "Celestial Mechanics", which is the mathematics of planetary orbits. At the end of his life, he decided to write a book summarizing all his discoveries. Unfortunately, he found that there were some results that he had proved many years ago, but now he had forgotten the proof, and couldn't reconstruct it. Whenever this happened, he dealt with the problem by writing, "It is easy to see that . . ."
"What immortal hand or eye,
could frame thy fearful symmetry?"
For years my favorite piece of data (taped to my office door) had those words scribbled on the bottom*. So I was never the smartest in my class, not even close. But I do wonder if (worry that) we don't support, challenge, encourage, our best and brightest kids enough. Or maybe that is best left to the family? (I was going to write that bright kids are a precious resource, but that just sounds insulting to everyone.)
*Far-IR spectroscopy of semiconductors, a symmetry in the electron and hole wavefunctions allowed us to designate the exciton absorptions we were observing... details not important.
Galois theory seems to be one of those subjects that lends itself to attractive exposition. I had a similar experience when I was the same age with Joseph Rotman's book. And, I, too, saw much of myself and my own failings in Galois' mercurial, intense persona.
What is the baseline familiarity/experience with mathematics necessary to make sense of this book?
In theory the book has an appendix containing all the necessary prerequisites, so somebody with good mathematical maturity would be able to just take it off the shelf and get going.
In practice, I'd recommend at least some familiarity with abstract algebra, especially field theory.
Artin's book on algebra is well-regarded, and available for free online (http://home.ustc.edu.cn/~liweiyu/documents/Algebra,%20Second%20Edition,%20Michael%20Artin.pdf). A lot of people love this book for its heavy emphasis on matrix groups and linear transformations, but I sort of hate it for the same reason (and in particular, he takes a few hundred pages to get to the material that's of relevance to Galois theory).
Dummit and Foote (https://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/0471433349) take more or less the opposite approach from Artin -- their work is quite abstract and dry, but encyclopedic, and introduces concepts in what I consider to be a sensible order. You should definitely not feel like you need to read all of this book to start on Cox -- the first few chapters, and then the material on field theory, should be more than sufficient.
If both of those options are too hard, the best really gentle introduction to abstract algebra that I know of is at the end of the second part of Harris and Hirst's book on discrete mathematics (https://www.amazon.com/Combinatorics-Graph-Theory-Undergraduate-Mathematics/dp/0387797106). They motivate the whole idea of group theory very nicely as part of a section on Polya's theory of counting, and the development is easy and natural.
For conceptual and philosophical material to hype you up on this journey, check out Hermann Weyl's "Symmetry" (https://www.amazon.com/Symmetry-Princeton-Science-Library-Hermann/dp/0691173257). Weyl is one of the 20th century's greatest mathematical physicists, and this is his book for laymen. He starts by analyzing different sorts of symmetry in nature and art, before gradually building up to the fully general abstract mathematical concept of symmetry that motivates group theory.
Thank you. I have very little real mathematical training or mathematical maturity. I read Euclid and Lobachevsky (and some Hilbert) at St. John's, but that's about all I have. I've been working through this book to start orienting myself: https://www.amazon.com/gp/product/0201102382
Fraleigh's book on Abstract Algebra--written for sophomore math majors--has a very nice section on Galois Theory.