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I am currently in an introductory Galois Theory course, and I thought it would be nice to compile a list of standard tricks for finding the Galois Groups of certain polynomials. I am studying from Stewart's Galois Theory, and while the book is very readable, he has given us one worked example (apparently the "canonical" $t^4-2$). Aside from some easy examples given in class, I feel like I should be more confident with these methods.

I'm aware there isn't one method for finding Galois Groups, but any tips or tricks, things to look out for, etc, would be much appreciated. Although I'm interested in all answers, keep in mind that this is an undergraduate introduction to Galois Theory, and thus far, we are only working over $\mathbb{C}$ (if you own Stewart's Galois Theory, my class is currently on Chapter 13). Also, I know that questions of this nature have been asked before, but often the responses link to papers / allude to topics not covered in an intro level galois theory course. Any help is appreciated!

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I think it's helpful to first consider the size of the Galois group, then ask which group of that size it could be. For example, if $|G|=6$, then it suffices to figure out whether $G$ is abelian –  Jay Kopper Oct 24 '12 at 19:54
    
See the books listed here. –  lhf Oct 24 '12 at 20:12
    
See also this question. –  lhf Oct 24 '12 at 20:13
    
I've started writing a mega-answer to this math.stackexchange.com/a/355012/448 –  David Speyer Apr 8 '13 at 19:08

1 Answer 1

up vote 5 down vote accepted

Apart from basic techniques, there are some beautiful theorems in mathematics, which can help you to find Galois group of a given polynomial over rationals. The theorem is called Dedekind and Frobenius Theorem which can be found in Dummit Foote Algebra book Chp. 13 and David Cox's Galois theory.

The theorem says that if you reduce an irreducible polynomial modulo primes not dividing the discriminant of the polynomial you get information about the elements of the Galois group.

Example: If g is a polynomial of degree 5. Reducing polynomial modulo some prime, if g remains irreducible, this means Galois group a 5-cycle. If g splits as quadratic times three linear polynomials. This means Galois group has an element of order 2, i.e., transposition. Therefore by a theorem in group theory you can conclude that G is S_5. There is also a probabilistic way of finding Galois group by using Chebotarev density theorem. However this probabilistic technique fails some times, such as in degree 8.

See the following for more details.

http://www.math.colostate.edu/~hulpke/paper/gov.pdf http://www.math.colostate.edu/~hulpke/talks/galoistalk.pdf

http://websites.math.leidenuniv.nl/algebra/Lenstra-Chebotarev.pdf http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisSnAn.pdf

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Oh I have just seen that you are under graduate, but you can still read these papers, you will not understand everything but you will understand methods to find Galois group. –  Reader Oct 29 '12 at 19:35
    
this is really wonderful. Thank you –  sperners lemma Oct 29 '12 at 19:35

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