History of Algebra: Modern Algebra, Ring Theory, Algebraic Topology

[Math Amateur]

I would really like to get a good understanding of how modern algebra developed, especially ring theory.

Two books I have recently bought are as follows:

Modern Algebra and the Rise of Mathematical Structures by Leo Corry.

and

Episodes in the History of Modern Algebra (1800 - 1950) by Jeremy Gray and Karen Parshall (Editors).

I have just started the book by Corry and it seems interesting.

I have skimmed the set of edited readings by gray and Parshall and it seems a bit daunting, but again interesting.

Does anyone else have books or online sources on the history of algebra?

I would also like to get some understanding of the history of algebraic topology ... but I had better learn some algebriac topology first ...:)

Peter

 

[Deveno]

The way ring theory developed was fairly "piece-meal".

By contrast, although certain examples of groups (like integers, probably the single most important group. EVAH!) were known from antiquity (other examples include symmetry groups and translational groups) and others became clear over time (both Euler and Gauss would probably recognize cyclic groups for what they are: additive congruences of the integers), it wasn't until the work of Galois and Abel that the essential abstract features of groups (such as homomorphisms, and normal subgroups) because clear enough to develop a cohesive theory.

After the works of Abel and Galois were published (both, tragically, died young) development was fairly rapid, by the end of the 19th century (just a half-century after Galois died, thereabouts) most of what students now learn in a first course on group theory was known.

Ring theory developed more slowly. Part of this is that rings have simultaneously "more" and "less" structure than groups: the multiplicative semi-group of a ring is not as "strict" a structure as a group, and thus there is less "intuition" to guide us. It is not immediately apparent what should take the place of "normal subgroup" in a ring, although EVERY additive sub-group is a normal additive sub-group, the right multiplicative condition took some time to arrive at.

The first steps towards what we would call "ring theory" began as problems in number theory: one useful tool in number theory is prime factorization. When the complex numbers became accepted as an extension field of the reals, it was hoped that similar constructions might provide answers to problems that were difficult to solve using integers alone (for an example of how this might be so, see my posts in mathbalarka's thread on http://mathhelpboards.com/number-theory-27/fermats-last-theorem-not-freak-just-special-one-8656.html#post40348).

This lead to the idea of "generalized integers" (for example, the Gaussian integers, or $\Bbb Z[\sqrt{2}]$). Unfortunately, the embedding of the integers in these "extensions" did not always preserve "primality". Kummer developed the theory of what we would now call "cyclotomic integers" (there has been some historical debate over whether his motivation was Fermat's Last Theorem, or, as is now common accepted, laws of higher reciprocity) obtained by adjoining complex roots of unity to the integers. He realized at some point that unique factorization did not hold in these structures, and sought "ideal numbers" that acted as "stand-ins" for the concept of prime number.

Dedekind sought to generalize this to algebraic number rings. Doing so, led him to the notion of an ideal. However, the general term of ring (or in German, Zahlring) did not come about until Hilbert, most likely as part of his attempt to put as much of mathematics as possible on axiomatic footing.

The last "big step" in the history of ring theory, was undoubtedly the work of Emma Noether, in my humble opinion one of the greatest mathematicians ever. She gave what are essentially the "modern definitions" of an ideal, and was undoubtedly the source for much of what became van der Waerden's classic Algebra.

One thing of note is that while normal subgroups of groups are themselves groups; in a commutative ring with unity, ideals aren't necessarily sub-rings (with unity) of these. Also, sub-rings aren't necessarily ideals, and this "wrinkle" tends to make for "messier theorems". It turned out that the "better" concept for clearing this up is that of a module (a term first used by Dedekind, I believe, but in a more limited sense than what we now use), and modules later became the prime example of what is now called an Abelian category (much of the development of category theory exists now just so that "Abelian category" makes sense: the idea is to prove something once in the categorical setting, replacing several independent proofs in individual settings). We have Ms. Noether to thank for that as well, who recognized that many structures could be thought of as modules over an appropriate ring (for example, any vector space can be made into an $F[x]$-module by considering a linear transformation $T \in \text{Hom}_{F}(V,V)$ and using the ring-homomorphism $F[x] \to F[T]$, which yields insights into such things as the Cayley-Hamilton theorem).

Anyway, in ring theory we have about 80 years from start to finish, compared with less than 50 as compared to groups. And most of this "gap" is because there are "more" useful properties to single out in integers as a ring, than in say, permutations as a group. Generalization occurred in "smaller steps", probably due to ring theory's deep roots in number theory (you have to give up "a lot of properties" to get a general ring: the division algorithm, principal ideals, greatest common divisors, unique factorization, cancellation, commutativity to name a few).

TL;DR version: look for a book that goes into detail about the works of:

Euler, Gauss, Kummer, Dedekind, Hilbert, and Noether (and Jacobson, if you want to know more about the development of "radical ideals"). There will undoubtedly be some others chronicled I didn't list here (Kronecker, Artin, Zariski, it's a long list...). That'll get you started.

Algebraic Topology is a LOT more recent, and to even get there, you'll want to at least know the basics of what a topology IS, and how the homology and homotopy groups came into being.

Topology is a bit abstract, most people first study the "baby example" of metric spaces, which builds directly on what they learn in calculus.

In any case, the subject is recent enough that a definitive history may not be possible, as the importance of newer perspectives in the long run remains to be seen. For example, one source I read: http://www.mathnet.or.kr/real/2009/3/McCleary_col.pdf barely mentions Weyl and certainly does not talk about Serre, Eilenberg, MacLane, or Grothendieck at all (it does talk at some length about Poincare and Brouwer).

(I know this isn't quite what you asked for. Oh well. Send me a bill).

 

[ThePerfectHacker]

You should learn topology first before learning algebraic topology. The two subjects are almost just as old, being from about 1900. Topology in its initial state felt more like set theory. While algebraic topology felt more geometric. This made Poincare' declare that topology is a disease that mathematics will have to recover from one day. He was wrong about that. Now it is standard to learn topology first before jumping into algebraic topology. What it basically is, is attaching algebraic structures to topological spaces. Then using algebraic machinery one can answer certain topological questions.

 

[Deveno]

You should learn topology first before learning algebraic topology. The two subjects are almost just as old, being from about 1900. Topology in its initial state felt more like set theory. While algebraic topology felt more geometric. This made Poincare' declare that topology is a disease that mathematics will have to recover from one day. He was wrong about that. Now it is standard to learn topology first before jumping into algebraic topology. What it basically is, is attaching algebraic structures to topological spaces. Then using algebraic machinery one can answer certain topological questions.

A favorite quote of mine by Hermann Weyl:

In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain

Of course, I would put it the other way around.

 

[Fantini]

I don't know algebraic topology, but I remember seeing a huge topic on MathOverflow about sypnosis for areas of mathematics, and it was something along the lines:

Algebra is hard, topology is easy, so...

 

[Math Amateur]

I don't know algebraic topology, but I remember seeing a huge topic on MathOverflow about sypnosis for areas of mathematics, and it was something along the lines:

Yes, maybe so ...

I would like to gain an understanding of homotopy and homology (both simplicial and singular) ... but as I have made no serious study of point set topology (I have a very rudimentary understanding of real analysis from many years ago ...) I have to estimate how much point set topology to do before beginning algebraic topology ... presumably I need a pretty good understanding of concepts like continuity, connectedness, and so on ... but I am not sure how much point set topology/analysis to do before beginning algebraic topology

Peter

 

[ThePerfectHacker]

You do not need to need to that much point-set topology to understand algebraic topology. A standard book on point-set topology has far more than what you actually need.

I find algebra easier than topology. I do not see any easy argument why $\mathbb{R}^n$ is topologically distinct from $\mathbb{R}^m$. But algebraically it is easy to show.