Wednesday, March 30, 2016

Algebra Too

Algebra seems to have had it's earliest roots in ancient Babylon, though many important ideas were contributed by Indian, Greek and other civilizations. The word itself is Arabic, from a word meaning "reunion of broken parts," and the subject was extensively developed by Arab mathematicians at the beginning of the second millennium. Most of us encountered some of the basic ideas, like solution of linear and quadratic equations in elementary school or (for old guys in benighted climes like me) high school. The most critical early contribution of the West was the invention of analytic geometry by Descartes and contemporaries. This unified the ancient streams of geometry and algebra, and was critical to the further development of nearly every area of mathematics. We got some of that in high school too.

Methods for solving linear and quadratic equations were known to the ancients, but early Western mathematicians* also found closed form solutions to cubic (powers to x^3) and quartic (powers to x^4) equations. The quintic (x^5) and higher powers stubbornly defied solution. The era of modern algebra really began when Abel and Galois proved that quintics and higher power equations could not in general be solved in terms of radicals (roots). In doing so they essentially invented the main subjects of what was called "Modern Algebra" in my college days: Groups, Rings, and Fields. Nowadays, books on such subjects are usually just called "Algebra" or "Abstract Algebra," not only because the work of a century and more ago hardly seems "Modern" anymore, but also because newer ideas now join the fray.

Anyway, when I found that I needed to learn a bit about rings I looked through my bookshelfs. The books I studied from long ago had either moved on to that great library in the sky, or, more likely, been tucked away in boxes somewhere. I found a few, including Contemporary Abstract Algebra by Joseph A. Gallian. I had a used copy, the early parts of which showed the dark marks of dust and skin oils, so somebody must have studied it. It was a third edition, another promising sign. I checked it out on Amazon and the current edition was the ninth, a sure sign of both success and author/publisher greed, as was the $200 asking price. The $8 I paid was clearly a bargain, and I only found one new chapter in edition 9, though even it was available in a much cheaper used 7th edition.

Abstract Algebra at Gallian's (elementary) level is studied by every undergraduate math major, a few confused physicists who think that they will learn the group theory needed for quantum mechanics (not bloody likely), a computer science major or two, and almost no one else. Looks like Gallian might have a good chunk of that market.

At the moment, it looks suitable for my purposes: elementary, short chapters, plentiful examples and exercises.

*The Persian Poet/Mathematician Omar Khayyam found a geometric solution to cubics in eleventh or twelfth century.