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I have been taking Abstract Algebra graduate classes, and we will begin studying Galois theory.

However, since it is a graduate-level course, the teacher has assumed everyone already had taken undergrauate-level algebra. Problem is, I come from an Engineering background, which did not have it.

Needless to say, I am a bit lost on what I need to know. The teacher will do a brief revision on fields, but I do not even know much about rings (I am fairly comfortable with groups though, since we studied it).

So, long story short, my question is: What do I absolutely must know about rings and fields to understand Galois theory?

I would also appreciate any references that may help. I have been reading Artin's Algebra, and Goodman's Algebra: Abstract and Concrete for now.


And, just to clarify, I do intend to learn more about Algebra later on. It is just that I need to keep up with the lectures, so I must learn the necessary things in a really short time.

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    $\begingroup$ Two pieces that you may be missing are A) "polynomial rings over a field". There aren't really many surprises there in the sense that largely it is about realizing that the things you are familiar with about polynomials over the reals work out the same over any coefficient field. You do need the basics about ideals of such polynomial rings. B) that linear algebra over any field works the same way as it does over the reals: matrices, systems of linear equations, ranks, dimensions, linear (in)dependence. This is often just glossed over (and it isn't difficult), but you do need to internalize it. $\endgroup$ – Jyrki Lahtonen Apr 30 '17 at 5:47
  • $\begingroup$ I would say in every case, abstract algebra and Galois theory is a very different subject to what you saw before. For the intuition, knowing well the characteristic and minimal polynomial of matrices will help. $\endgroup$ – reuns Apr 30 '17 at 6:43
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    $\begingroup$ You also need to have some knowledge of groups apart from polynomial rings and fields and this part I find more difficult than the polynomial rings or fields. In particular one needs to be familiar with cyclic groups, abelian groups, permutations groups, normal subgroups and famous theorems of Cauchy/Langrange/Sylow. Since you have studied this I hope you will be able make fruitful use of this when studying Galois theory. $\endgroup$ – Paramanand Singh Apr 30 '17 at 8:09
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    $\begingroup$ I think you're better off asking the teacher than us random strangers. $\endgroup$ – Gerry Myerson Apr 30 '17 at 12:55
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    $\begingroup$ My advice (for what it's worth-which may be exactly what you pay for it): concentrate on the group $S_4$ and all its subgroups in detail (including those "in disguise"). Secondly, recognize the Fundamental Isomorphism Theorem for rings is used to identify extension fields as quotient rings of a polynomial ring, and that an extension ring (or field) of a field is also a vector space over that base field, and linear algebra thus applies. $\endgroup$ – David Wheeler Apr 30 '17 at 15:44
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OK, I think my question might be wandering in the not-answered list of questions, so I'll share my thoughts.

A good book I've been using is Gallian's Contemporary Abstract Algebra. It is very well organized, and has a good number of examples and exercises (some solved). Irrelevant to the actual content, but I also liked the biographies of some mathematicians, which can be an inspirational breather.


From what I've seen, you need some knowledge about rings and especially ideals. Integral domains and unique factorization domains play some roles here and there. Also, a very strong knowledge and familiarity of fields and extension fields is highly desirable.

Group theory is also important, but I've been studying these a lot (even if only because of a test).

Specific topics you want to be familiar with are the symmetric group and especially polynomial rings. This last one can't be emphasized enough, in my opinion, but I think it suffices to say that they were the main motivation for the development of Galois Theory.

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