# Gracious Living

Ideals: Numbers But Better
March 8, 2011, 14:27
Filed under: Algebra, Math | Tags: , , ,

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.

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Rings, Integral Domains, and Fields
February 2, 2011, 19:36
Filed under: Algebra, Math | Tags: , , , , , ,

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 $\mathbb{Z}$ and $\mathbb{Q}$ similarly to how the Peano axioms constructed $\mathbb{N}$.