MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am interested in topology, while I am not so comfortable with some algebraic flavored textbook on algebra. Actually, it was not until I learned some topology that I began to understand some abstract algebra. I believe that behind every algebraic theorem, there is a geometric analogue, and this is what I am interested in.

I want to find algebraic textbooks that are of geometric flavor. Say, Armstrong's book, Group and Symmetry is a lovely book on group theory that is of this type. Are there some more such textbooks (on ring theory and homological algebra)?

Thank you!

share|cite|improve this question
Does algebraic geometry count for the geometric flavor you're looking for? – Fernando Muro Jan 5 '15 at 9:23

Confusingly, in addition to Emil Artin's Geometric Algebra mentioned by Anweshi, there is Michael Artin's textbook Algebra. The latter is more of a modern first-year graduate textbook covering the usual topics you'd find in a book like Lang, Dummit and Foote, and so on. I don't have Michael Artin's book handy but I remember a lot of it having a "geometric flavor," e.g. there was a lot of discussion of symmetry groups and linear representations.

This was one of the first abstract algebra books I read and I remember loving it at the time.

share|cite|improve this answer
Couldn't have agreed more! – Somnath Basu Jul 27 '10 at 3:55
Michael Artin's book is a nice read, but I wouldn't quite put it at the first-year graduate level. – Elizabeth Henning Jan 5 '15 at 7:55

I guess it depends what you consider geometry. I think Eisenbud's book Commutative Algebra with a View Towards Algebraic Geometry is magnificent, and motivates commutative algebra from stem to stern by means of the connections with geometry. I wish it had existed when I was a graduate student. But it's algebraic geometry, not algebraic topology, for sure.

share|cite|improve this answer
I think this should be taken with a grain of salt. A lot of the motivation in Eisenbud may only be "magnificent" after you have some understanding of the algebraic geometry involved. – Charles Staats Jul 27 '10 at 15:13

Try Artin's Geometric Algebra.

You might also find interesting Shafarevich's book “Basic Notions of Algebra”, for understanding the philosophy behind algebra. I strongly recommend that you spend some time with that book.

Also try Coxeter's various books such as Projective Geometry, etc..

But of course if you really want to understand the "geometry behind the algebra", then you should first have a good footing in algebra, and then you should study algebraic geometry. Even for Artin's Geometric Algebra, you better first have an understanding of algebra before reading it. It appears to me that at the moment the best bet for you will be Shafarevich's "Basic Notions of Algebra".

share|cite|improve this answer
Oh my god, "Basic Notions of Algebra" is fantastic! I always thought that this type of book does not exist yet and I already started to write something like that :-). – Martin Brandenburg Jul 27 '10 at 7:13
@Anweshi: Shafarevich is the only author of the book “Basic Notions of Algebra”. – Dmitri Pavlov Jul 27 '10 at 11:08
@Dmitiri Pavlov: Such was my feeling too. But before making the answer I looked in Amazon and saw this:… – Anweshi Jul 27 '10 at 12:59
@Martin: :-) ... – Anweshi Jul 27 '10 at 13:02
Dmitri is right; Kostrikin is one of the editors of the series the book appeared in. Speaking of Kostrikin, he coauthored "Linear Algebra and Geometry" with Manin. – Franz Lemmermeyer Jul 27 '10 at 13:21

One fundamental aspect of the connection between geometry/topology and algebra is the analogy between galois theory of fields and covering theory in algebraic topology and algebraic geometry. A nice book on the subject has appeared recently : Tamas Szamuely's Galois Groups and Fundamental Groups.

share|cite|improve this answer

You may also enjoy "Visual Group Theory" by Nathan Carter.

share|cite|improve this answer
There's also the associated software, Group Explorer, at It can be used independently of the book. – J W Jul 27 '10 at 10:36

Try Transformation groups for beginners by Duzhin. It's group theory, motivated by geometry.

share|cite|improve this answer

manin's linear algebra and geometry

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.