I left you with a bit of a teaser. We’d defined rings, integral domains, and fields, and even seen a few examples, but in such a short exposition, there wasn’t very much time to give you the tools to work with them. There turn out to be ideas that make better sense in a ring, like primality and divisibility. But to understand them, we need to develop a little machinery, which in this case is the theory of ideals. As I show below, ideals are like better-behaved numbers, and help us understand the structure of, among other things, the integers.

Filed under: Algebra, Math | Tags: abelian groups, algebra, arithmetic, field theory, group theory, Math, ring theory

In which I sort of breeze through a couple of really awesome and really important concepts. Last time, we classified abelian groups — now we’ll see what happens if we require additional structure on the groups. In particular, I’m going to construct and similarly to how the Peano axioms constructed .