How we can use algebra
James B. Wilson
This work is licensed under a Creative Commons Attribution 4.0 International License.
I learned algebra as an undergrad in Noether styled "Groups, Rings, and Fields". At the same time I was a software technician at Intel writing Object Oriented Programs. Combining both influences I created Java programs for all the objects placing them in an interconnected heirarchy. I noticed gaps between possible axiom combinations and dug deeper, often inventing new algebras, or at least I didn't know about them from books and the internet at the time was thin on those topics. I then set out to do all the exercises in Hungerford's graduate algebra text, some of my solutions can still be found on the web, though all of those copies you might find are violating my copyright and I never wanted them shared. In grad school I learned the second round of algebra, still groups, rings, and fields, but now with structure theorems and a modest amount of organization around categories. Again I solved all the exercises in the algebra course and the hundreds in the 50 years of old qualifier courses. I also kept computing earning my Ph.D. in computational algebra. But along the way I got stuck solving problems such as how do you find a direct product in a group, computationally fast? At the critical stage the problem boiled down to p-groups, then nilpotent Lie algebras which most of us knew to exist but no one seemed to understand, then bilinear maps, then... an algebra that had the wrong minus sign to be right! It was then in frustration that I showed my advisor Bill Kantor and he said:
You know there are other algebras, like Jordan algebras.
And there it was. An entire world of algebras non-associative but in the charming words of the Russian school ``Nearly associative''. It broke the problem. It solved a 40 year old conjecture. It opened the door to numerous advances in computation and algebra within my field. And it was all over a minus sign. Stunningly easy when you know to look. And why did it take so long after all the preparation I had done solving exercise from every small corner?
Because algebra isn't "Groups, Rings, and Fields".
This book lays out a method to learn algebra the focusses on unified concepts along side outward facing goals. So to each abstraction there should be a corresponding practical consideration. I believe this honors Noether even more than can be said from emphasizing Groups, Rings, and Fields. Her own work spanned foundations, algebra, abstraction, and physics. With the goals set forth the atoms of algebra becomes the operators. Groups become misbehaving rings. Linear algebra takes over in solving problems but also inspires structure. And algebra viewed so librarly appears everywhere from designing software packages, to classifying quantum particles, to recognizing clusters in high-dimensional data.
You don't have to give up on Groups, Rings, and Fields to use these notes. This is just my window into the world of algebra. I hope it will give you a new way to look in on this beautiful subject.