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All:

I had studied abstract algebra long time ago. Now, I would like to review some material, particularly about Galois theory (and its application).

Can anyone recommend an abstract algebra book which cover Galois theory (and its applications)?

I have been a software engineer for past many years. Ideally, I would like an algebra book with programming assignments or exercise (help me to understand concept.) For example, homework assignments to write a program to verify Galois theory, or to construct 'solvable Group", or anything like that. (Of course, I can think some random questions, but I prefer a text book with well designed, and meaningful home works).

I felt that I was not good at deriving formula anymore, I would like to use my programming skills to help me to understand subject, do more hand-on exercise and calculations.

Olexandr Konovalov
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Ben
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    Interesting question. Abstract algebra seems like a subject that could lend itself well to work involving computer software. Here's an exercise for you. Suppose you know that $a$ and $b$ are roots of two given polynomials whose coefficients are integers. Write a program to find the minimal polynomial of $a+b$. – Michael Hardy Aug 10 '14 at 21:18
  • That's not a well-posed problem, Michael. :) Ask for some polynomial with root $a+b$. – darij grinberg Aug 10 '14 at 21:19
  • ....um... @darijgrinberg : Do you mean the answer might depend on which of the roots of the two given polynomials are chosen? ${}\qquad{}$ – Michael Hardy Aug 10 '14 at 21:46
  • @darijgrinberg : I suppose what I meant by minimal polynomial is a monic polynomial with rational coefficients having $a+b$ are a root. There's only one of those (provided $a+b$ is well defined). ${}\qquad{}$ – Michael Hardy Aug 10 '14 at 21:49
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    While it doesn't cover Galois theory, Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea (http://www.springer.com/mathematics/algebra/book/978-0-387-35650-1) is an accessible intro to commutative algebra and algebraic geometry with an emphasis on writing programs for problem-solving. – Gyu Eun Lee Aug 10 '14 at 22:03
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    In addition to Ideals, Varieties and Algorithms, I came here to suggest Abstract Algebra with GAP. Sadly, it concludes with Galois theory. http://college.cengage.com/mathematics/gallian/abstract_algebra/6e/shared/gap/full_manual.pdf – dls Aug 10 '14 at 22:42
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    Two more links: http://www.math.colostate.edu/~hulpke/CGT/howtogap.pdf http://www.gap-system.org/Doc/Teaching/teaching.html – dls Aug 10 '14 at 22:46
  • @MichaelHardy: Yes, it depends on the choice of the roots. For a stupid example, let $a$ and $b$ be roots of the same irreducible polynomial. – darij grinberg Aug 10 '14 at 23:01
  • It is not an algebra book per se, but you may like at least parts of Modern Computer Algebra by J. von zur Gathen & J. Gerhard. It may be a bit more advanced than what you hope at times. At least to an extent a reference book rather than a book for learning the basics, so browse before you shop. A lot of pseudocode in there. – Jyrki Lahtonen Aug 11 '14 at 12:30
  • Another suggestion - Abstract Algebra: An Interactive Approach by William Paulsen. – Olexandr Konovalov Aug 11 '14 at 14:46

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All books listed in the comments above are listed in my answer to the question Novel approaches to elementary number theory and abstract algebra, so I am placing a CW-answer here to remove this question from the unanswered queue.

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Olexandr Konovalov
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