Topics in Commutative Ring Theory
Chapter One
Rings and Subrings
The Notion of a Ring
In 1888 - when he was only 26 years old - David Hilbert stunned
the mathematical world by solving the main outstanding problem in
what was then called invariant theory. The question that Hilbert settled
had become known as Gordan's Problem, for it was Paul Gordan who, 20
years earlier, had shown that binary forms have a finite basis. Gordan's
proof was long and laboriously computational; there seemed little hope
of extending it to ternary forms, and even less of going beyond. We
will not take the time here to explore any of the details of Gordan's
problem or even the nature of invariant theory (and you shouldn't be at
all concerned if you don't have the foggiest idea what binary or ternary
forms are or what a basis is), but Hilbert - in a single brilliant stroke - proved
that there is in fact a finite basis for all invariants, no matter how
high the degree.
The structure of Hilbert's proof is really quite simple and is worth
looking at (again we will not worry at all ab ... read full excerpt from Topics in Commutative Ring Theory ebook
