# 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.

The Construction of the Reals: Metric Completion and Dedekind Cuts
March 6, 2011, 21:31
Filed under: Algebra, Math, Set Theory, Topology | Tags: , , , ,

It looks like I’m getting views now, which is surprising.  I’ve been pretty busy with schoolwork, but I really want to get this blog up to speed, particularly because I’d like to start discussing things as I’m learning about them.  I’d also like to make more non-mathematical posts, but maybe these are best left to a separate blog?  Thoughts?

Our first example of a field was the field of rationals, $\mathbb{Q}$.  Recall that this was the field of fractions of the integers, which were in turn the free abelian group on one generator with their natural multiplication.  But now it appears that we’re stuck.  While we intuitively know what $\mathbb{R}$ should be — it’s a line, for crying out loud — there seems to be no algebraic way of “deriving” it from $\mathbb{Q}$.  A first guess might be to add in solutions of polynomials, like $\sqrt{2}$ as the solution of $f(x)=x^2-2$, but not only does this include some complex numbers, it also misses some real numbers like $e$ and $\pi$.  (We call such numbers — those that aren’t solutions of polynomials with rational coefficients — transcendental.  It’s actually quite difficult to prove that transcendental numbers even exist.)

Instead, we turn to topology.  Below, I give two ways of canonically defining $\mathbb{R}$, one using the metric properties of $\mathbb{Q}$, one using its order properties.  I found this really interesting when I first saw it, but I can’t see it interesting everyone, so be warned if you’re not a fan of set theory or canonical constructions.  One of the topological techniques we’ll see will be useful later, but at that point it’ll be treated in its own right.

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 $\mathbb{N}$.

The Classification Theorem for Finitely Generated Abelian Groups
January 29, 2011, 16:23
Filed under: Algebra, Math | Tags: , , , ,

Wow, it’s been a long time since I’ve written anything on this blog.  I’m taking algebraic topology and an algebraic number theory course this semester, and I started reading through Atiyah and MacDonald’s Commutative Algebra over the winter.  So I thought I’d continue with a little algebra.  The algebra we’ve done thus far has been highly noncommutative, for the most part — we investigated groups like free groups, symmetric groups, matrix groups, and dihedral groups in which the order of operations mattered.  As you might expect, with abelian groups, the theory becomes much simpler, and the subject called “commutative algebra” is just the study of abelian groups with extra structure — something like a scalar multiplication, as in the case of vector spaces, or some other operation.  But first, we need to understand abelian groups.

When talking about abelian groups specifically, we usually write them additively: the group operation applied to $a$ and $b$ is $a+b$, and then we can build expressions like $3a+2b$.  The proof I give below is due to J. S. Milne, who in turn says it’s similar to Kronecker’s original proof.  Of course, I’ve added more detail in places where I thought it was necessary, and taken it out where I thought it wasn’t.  There are other, more common proofs, typically using matrices, but I find them unwieldy and inelegant.

Countability Axioms
December 31, 2010, 01:55
Filed under: Math, Topology | Tags: , , , ,

So far we’ve seen two basic families of properties of topological spaces.  The connectedness axioms tell us in what ways it is possible to break our space apart into pieces.  The compactness axioms tell us how bounded the space is.  What we’re going to look at today is a set of axioms that deal with cardinality.  It should be mentioned that topology, for the most part, doesn’t really care about large cardinals — at most, we’re dealing with $\aleph_1$, the cardinality of our favorite counterexample $\omega_1$, and $c$, the cardinality of the reals.  These are equal if we accept the continuum hypothesis, and in either case we often talk about them in terms of countable subsets — sequences and the like.  The reason that countability is so important is that the properties we’re about to study are typical of metric spaces, and metrizability is a central question of point-set topology.

Limit Point and Sequential Compactness
December 30, 2010, 02:51
Filed under: Math, Topology | Tags: , , ,

Back to topology.  The interesting thing about compactness, as I see it, is that its definition isn’t very intuitive.  We want to talk about what are basically “closed and bounded” sets without really using closedness, which doesn’t behave well with subspaces, or boundedness, which doesn’t behave well with anything.  At the time this idea came about in the early part of this century, the mathematicians ssewho invented it (Borel, Weierstrass, and Bolzano, mainly, I think) messed around with a couple other definitions but they turned out to rely on intuition from Euclidean space, and so not be nearly as useful.  In this post, we’ll study those other definitions.

I debated with myself for a long time about what to do first: this, the separation axioms (which measure how good the topology is at distinguishing disjoint sets), or the countability axioms (which measure, well, how countable the topology is).  In a sense, each one depends on the other two for examples and theorems, and in this case, though it’s easy to show that all the properties I’m listing are different, we’ll need to refer to the other properties to discover when they are the same.  So I expect this post to be brief; after we have all the other properties defined, it’ll be easier to talk about what implies what.

Subgroups of Free Groups are Free
December 28, 2010, 07:32
Filed under: Algebra, Math | Tags: , , , , ,

Okay, first post for a while.  As I promised quite a while back, let’s prove together that subgroups of free groups are free.  It’s surprising that this is nontrivial to prove: just try to come up with some subgroups of $F_2$ and you’ll see what I mean.  In fact, using only basic algebraic topology and a bit of graph theory, we can come up with a really simple argument that replaces this one.  Perhaps that’s an argument in favor of algebraic topology.  But I think this angle is sort of interesting, and it should be a fresh experience for me, at least.

The proof is due to Jean-Pierre “Duh Bear” Serre in his book Trees.  A heads up if you track this down — Serre has a really weird way of defining graphs.  Fortunately, for this proof at least, a little bit of work translates things into the same language of graphs and digraphs that we saw when talking about Cayley graphs.  I review that below the fold.  It takes a while to set up the machinery, though the proof itself isn’t too long.  To recompense, I’ve left out a couple minor details, which you’re probably able to fill in.  If some step doesn’t make sense, work it out — or try to disprove it!