The most common theorems taught in Abstract Algebra
I am self learning abstract algebra. I want to know which theorems are a must to understand.
Now these are limits I have to deal with (please consider when answering):
- I have limited internet access
- Few mathematical books written in English are available.
- I can not afford to order books from abroad．
I just want to know what is the core knowledge (theorems, lemmas, etc) of any decent graduate level abstract algebra class.
Solution 1:
Assuming you are aiming for a Groups/Rings/Fields, at a level one might expect from someone who has taken a year's undergraduate sequence but not much more:
Groups
- Definition, lots of examples.
- Subgroups, homomorphisms, normal subgroups
- Cosets; Lagrange's Theorem.
- Group actions.
- Permutation groups. Symmetric and alternating groups.
- Cayley's Theorem.
- The Isomorphism Theorems.
- Cauchy's Theorem and the Sylow Theorems.
- Abelian groups.
- The Fundamental Theorem of Finitely Generated Abelian Groups.
Rings
- Definition, lots of examples.
- Homomorphisms.
- Ideals.
- The Isomorphism Theorems.
- Field of quotients of integral domains.
- Polynomial rings.
- Euclidean rings, Principal Ideal Domains, Unique Factorization Domains
- Gauss's Lemma and polynomials over Unique Factorization Domains.
Fields
- Field extensions.
- Algebraic extensions. Dedekind's Product Theorem.
- Primitive Element Theorem.
- Separability.
- Fundamental Theorem of Galois Theory (finite case).
- Solvability by radicals.
- Finite fields.
I think Herstein's Topics in Algebra is a good resource, though it is a bit old-fashioned (it has almost nothing on group actions, for example).
With a more time, I would add the following topics:
Groups
- Introduction to Semigroups.
- Semidirect products; extensions.
- Jordan-Höder Theorem.
- Commutator subgroup; solvability (this will have been covered somewhat when discussing solvability by radicals under Fields; study it in more detail).
- Basic results on $p$-groups.
- Divisible and reduced abelian groups.
- Structure Theorem of Divisible Abelian Groups.
- Free groups, free products, free products with amalgamation.
- Basics of representation theory of groups.
Rings
- Jacobson radical.
- Wedderburn's Theorem.
- Wedderburn-Artin Theory.
- Division rings.
- Localizations.
- Group rings.
- Basic theory of modules.
- Modules over PIDs.
Fields
- Inseparable extensions.
- Transcendental extensions.
- Kummer Theory.
- Normal Basis theorem.
- Fundamental theorem of Galois Theory (infinite case).
Lang's Algebra is a good resource, but is not easy to learn from. For Ring Theory, Lam's First Course in Noncommutative Ring Theory is very good.
Solution 2:
You leave quite a few restrictions. You don't have books, can't get books, and can't use the internet much. So, there isn't really much for us to tell you. You HAVE to have some form of book. You can't learn from nothing but a list of a few theorems. So, it seems the least restrictive of your list is the internet. Obviously, you can use it some.
By the way, you can't just learn the most important theorems. The rest of the stuff in the book leads up to these theorems and the theorems make no sense, or can not be proven, unless you have all the rest.
This should get you a lot of the basics. But, it's on the internet. As I said before, you basically left no possibilities for help, so I have to delete one of your restrictions.
http://www.math.niu.edu/~beachy/abstract_algebra/study_guide/contents.html
Other than that, search for undergraduate abstract algebra online. "Intro" or "elementary" are words that tell you it's going to be undergrad level. Here are lecture notes from Hungerford's undergrad book, which is the book I used as an undergrad. If you could at least print this off, or something like it, that would solve the internet access problem.
http://www.math.msu.edu/~meier/Classnotes/M411S06/M411S06notes.pdf
Solution 3:
Warning.
I am an undergraduate student in Math. I have no idea whatsoever about a graduate level Abstract Algebra course, but, I thought I'll sketch a list of resources one should master before graduate School (in my opinion).
Group Theory:
- Group action (with as many examples and applications as you can).
- Sylow's Theorems
- The concept of semidirect product (with as many examples, and it's use in construction of groups)
- Lots of examples of groups ($M_n$, $GL_n(\mathbb C)$, $GL_n(\mathbb F_q)$ $SL_n$, $PSL_n$, etc). By this, I mean, as many theorems and non-trivial facts about these groups as you can.
- Permutation groups (inside out)
- Free groups
- Solvable groups (Hall's Theorem), Nilpotent group, Central series
Where Should I learn these from?
Well, this is a question to which there is no right answer always. So, here is my choice not in order .
- Dummit and Foote (The Modern encyclopedia of Abstract Algebra)
- M.Hall -Theory of Groups
- M.Aschbacher-Finite Group Theory
I would like to emphasize that you must have good command over examples and counter examples for exceptions, theorems (in which some hypothesis plays a crucial role) and so on, either from books or through self-Construction.
(P.S.: This is an advice I received from a Senior of mine who is now a Ph.D student and I have edited it suitably (for the forum) and added my own experience through the learning process.)
For other branches of Abstract Algebra, I'd write it up as I get time. Please bear with me!