Integral Domains and the failure of unique factorization

Introduction

Recall the Venn diagram illustrating special kinds of integral domains.

I want to look at integral domains in general, but integral domains that are not unique factorization domains (UFDs) in particular. I’m interested in the outer ring of that diagram.
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Introduction to Rings

Edit 7/12/12. Write Z[(1+\sqrt{-19})/2]\ , near the end. See the second “edit”.
Edit 7/17/12. Write a + (b/2)(1+\sqrt{D})\ near the beginning… so this is the earliest “edit” Life would be so much simpler if all quadratic integer rings looked like a + b \sqrt{D}\ , but they don’t! We’ll get to this in the next post.

The material I’m about to introduce comes from a 1st undergraduate course typically called abstract or modern algebra.

What I really want to talk about are number systems of the form

a + b \sqrt{D}

where D is an integer, negative or positive (and in some cases numbers of the form \frac{a + b \sqrt{D}}{2}\ – oops, edit: make that a + (b/2)(1+\sqrt{D})\ ). These are called quadratic integer rings. What fascinates me about them is that they have many of the properties of the integers – but at least some of them lack the most fundamental property of the integers.

The fundamental theorem of arithmetic is the one that says that any integer can be written essentially uniquely as a product of primes.

Consider, however, the products

(1+2\sqrt{-5})(1-2\sqrt{-5} = 3\cdot7 = 21\ .

That is a big deal – because in the system of numbers of the form

Z[\sqrt{-5}] = a + b \sqrt{-5}\ ,

all 4 of those numbers are irreducible: they themselves cannot be factored.
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Group Theory – Building abelian groups

I propose to find all the abelian groups up to order 100. It’s pretty easy, and there’s one nice idea that will simplify things. An abelian group, I hope you recall, is one that is commutative.

(The smallest non-abelian group is D3, the dihedral group of order 6… which is isomorphic to S3, the symmetric group on 3 symbols. All other dihedral groups are non-abelian, and the quaternion group of order 8 is non-abelian. But we’re going to look for abelian groups.)
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Group Theory – Direct Products

One of the ways of producing (potentially) new groups from old is called the direct product.Let me show you the direct product of two groups by example, first.

Let me invoke the abstract algebra package and set it to “groups”.

C2 x C2

I want two copies of C2, the cyclic group of order 2…

… and I am going to ask for their “direct product”.

What did we get?

First of all, we see that the elements of the direct product are just ordered pairs – the first component from H, the second component from K.


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Group Theory: normal subgroups

I want to show you a simple concept. It has major consequences, but it starts out as a far simpler thing than it usually looks like.

I want to talk about normal subgroups. Clearly, if all subgroups were normal, we wouldn’t need to distinguish normal from non-normal… so I will have to show you a non-normal subgroup. That won’t be a problem.

Before I go into the details, let me give you the answer. Even if you know this material, the answer may surprise you.

A subgroup K of a group G is a normal subgroup of G if and only if K is the kernel of a homomorphism defined on G.

I know… I know: besides “normal”, I’ve used three other terms that I haven’t defined: subgroup, kernel, and homomorphism. If you’ve never seen this material before, you’re lost – but I can fix that.

Again, before I go into the details, let me give you the names of other things I will be talking about: fibers, left cosets and right cosets, and factor groups; I will briefly mention conjugacy classes.

My starting point will be to define a (group) homomorphism. It is just a function between groups which honors the group multiplications. (They do not need to be, and generally are not, the same operation in the 2 groups.)
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Group Theory: Dihedral Groups

Having looked at cyclic groups, let’s look at dihedral groups. They can be created geometrically by starting with a cyclic group Cn… think of it as rotations of a regular n-gon through multiples of 360°/n… and then imagine that you can also spin the n-gon out of the plane about some axis.

Huh?

Let’s look back at the cyclic group C3. Take an equilateral triangle. Take our group operation r to be a rotation thru 120° (= 360°/3). Then r^2 is a rotation thru 240°, and r^3 is a rotation thru 360° – that is, r^3 = 1, the identity element.

Here a pictures of what r and r^2 do to an equilateral triangle:


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Introduction to Group Theory and Cyclic Groups

My goal in the next few posts is to talk about low order finite groups – that is, groups which contain a small number of elements.

My introduction to groups is going to be rather nonstandard. And it will be sketchy. Grab your favorite Introduction to Abstract Algebra or Introduction to Group Theory book. Suggestions:

  • Fraleigh, “A First Course in Abstract Algebra”. A popular introductory text. I own two different editions.
  • Dean, “Classical Abstract Algebra”, ISBN 0060416017. An excellent, if little known, introductory text.
  • Dummit & Foote, “Abstract Algebra”. Written for undergraduates, with enough material for grad students. This was my main reading this time around.
  • Schaum’s Outline of Group Theory. Cheap.
  • Armstrong, “Groups and Symmetry”, ISBN 0387966757. An excellent undergraduate text with emphasis on the actions of groups on geometric figures. I’ve been using this book, too.

I think the cleanest starting point is to define a group as follows. We start with a set G, and a binary operation * on it. That is, given any two elements a, b of the set G, we have the product a*b, and that product is an element of G. Specifically, we require that the set be closed under the operation (or product). (That rules out, for example, the dot product of two vectors – because the dot product is not a vector, but a scalar.)
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