# Historical textbook on group theory/algebra

Recently I have started reading about some of the history of mathematics in order to better understand things.

A lot of ideas in algebra come from trying to understand the problem of finding solutions to polynomials in terms of radicals, which is solved by the Abel-Ruffini theorem and Galois theory.

I was wondering if there's a textbook (or history book which emphasizes the mathematics) that goes vaguely in chronological order or explicitly presents theorems and concepts in their historical context.

Alternatively, if you think it would be better to attempt to read the original papers (Abel's famous quote about reading the masters comes to mind), such as the works of the mathematicians mentioned in this wikipedia article , how would I go about doing so?

EDIT: while researching some of the recommended books, I found this interesting list (pdf) of references.

The inspiration for this question came from asking myself "why would someone bother/think of defining a normal subgroup in the first place?" (although I already know the answer) and hence I am asking about Galois theory, but really this question works for any area of mathematics and perhaps someone should open a community wiki question for that.

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See van der Waerden's "A History of Algebra" and J. Nicholson "The Development and Understanding of the Concept of Quotient Group" (Historia Mathematica 20 (1993), 68--88), and B.M. Kiernan "The Development of Galois Theory from Lagrange to Artin" (Arch. Hist. Exact Sciences 8 (1971), 40--154). – KCd Jan 9 '11 at 6:51

I found the book The genesis of the abstract group concept, by Hans Wussing, to be very interesting. It gives a scholarly history of the development of the concept of group, from its roots in number theory, geometry, and the theory of equations prior to the 19th century, through to (more-or-less) the end of the 19th century.

As for reading the original papers, while there is a lot to be said for this, you should bear in mind that these are not only typically written in languages other than English, but are also in a mathematical language that is quite different from our modern language. If you do decide to look at the originals, Wussing's book would be a good guide to the major papers in the initial development of group theory.

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Please allow me to quote from Joseph Rotman's book "Galois Theory", as follows

"When writing the definition of Galois group for this text, I asked myself an obvious question: how did such thoughts occur to Galois in the late 1820's? The answer, of course, is that he did not think in such terms; for its first century, 1830-1930, the Galois group was a froup of permutations. In the late 1920's, E.Artin, developing ideas of E.Noether going back at least to Dedekind, recognized that it is both more elegant and more fruitful to describe Galois groups in terms of field automorphisms(Artin's version is isomorphic to the original version). In 1930, van der Waerden incorporated much of Artin's viewpoint into his influential text "Moderne Algebra," and a decade later Artin published his own lectures. So successful have Artin's ideas proved to be that they have vertually eclipsed earlier expositions. But we have lost the inevitability of the definition; group theory is imposed on the study of polynomials rather arising naturally from it. This appendix is an attempt to remedy this pedagogical problem by telling the story of what happened in the beginning. The reader interested in a more thorough account may read [Edwards] or [Tignol] "

H.M.Edwards, Galois Theory,Springer,1984

J.-P.Tignol, Galois Theory of Algenraic Euqations, Wiley, 1988

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Unknown Quantity by John Derbyshire is an excellent historical account of the evolution of abstract algebra. The book is aimed at the "mathematically inclined" and covers the history of algebra from birth to present day (or pretty close.)

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Algebra is a very large field, so you probably want to be a bit more specific.

In case you are wondering about Galois Theory, and want to learn its history for the purpose of understanding it, I strongly recommend the following book:

Jörg Bewersdorff. Galois Theory for Beginners. AMS 2006.

It doesn't present the history in a scholarly fashion, but as a means to the end of understanding the modern formulation of Galois Theory in terms of fields and automorphism groups. I don't think I would have understood any of it without this book.

There is also a paper

John Stillwell. Galois Theory for Beginners. The American Mathematical Monthly. Vol. 101, No. 1 (Jan., 1994), pp. 22-27.

with a similar goal. I'm mentioning it because the author, John Stillwell, has written another marvelous book about the history of mathematics in general.

If you are interested in Abel's concurrent proof of the unsolvability of the quintic, have a look at the book

Peter Pesic. Abel's Proof. MIT Press 2004.

although it's written for the general public. However, it does include a modern transcription of Abel's original proof in an appendix, which is the object of interest here.

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yes, sorry. I should have made it more clear that this question was asking about Galois theory and the related ideas. – Harry Stern Jan 8 '11 at 23:37
Also, I accepted Matt's answer (I had a really hard time choosing, actually, as I haven't actually read either books yet :) but I really appreciate yours as well and will be looking into the books you mentioned. – Harry Stern Jan 8 '11 at 23:47

I highly recommend RBJT Allenby's "Rings, Fields and Groups" for giving the both the historic development of modern algebra and biographic sketches of many of the mathematicians involved. He starts of with polynomial equations then follows how that lead to generalizations of rings and ideals, field theory, groups and Galois theory.

Armstrong's "Basic Topology" does a similar approach for Topology starting with polyhedra in Euclidean space and moving to open sets, topologies, triangulations and homotopy groups.

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