01.04.10 / grad school, math in english, real deal math / Author: JHill / Comments: (0)
When I tell people I do algebra, they often reply, “I was good at algebra in high school. How hard can it possibly get at the graduate student level?” Well, I have decided to finally answer this question with something I find REALLY WEIRD while also demonstrating the complexity of the subject.
I have designed this entry so that all you need to know is basic high school algebra. I will explain everything else you need to know (but feel free to ask questions). Now, it’s going to take me while to get to the punch line, so please just bare with me as I go through the background material.
(I should also say, if you are a legit mathematician then this isn’t the post for you. I have simplified alot of things to make this entry reader friendly, but please let me know if I made an heinous errors.)
1. Integers, Primes and the Fundamental Theorem of Arithmetic
Integers were called “whole numbers” when I was in high school. So, these are numbers like -1, 0, 1, 2, 100, etc.
Primes are integers that can only be divided by 1 and itself, like 2, 3, 7, 127.
Every number has a prime factorization. This means that you can “break down” every number until you are just multiplying a bunch of primes.
Examples are:
It’s just like making “factoring trees” when you were in high school. If you have a prime number, then the only factor is itself. The Fundamental Theorem of Arithmetic states that the prime factorization for any number is unique (except for the order that you multiply the numbers). That means that the only way to factor 10 is 2 x 5, and the only way to factor 3 is 3. This theorem is the first thing you learn in an elementary number theory course, or it comes back when you are looking at factorization in Modern Algebra.
2. Rings
Now here’s where things get a little more complicated. Ring Theory is one of the major topics covered in Modern Algebra, and you are going to have to understand what this is before we can move on.
It might be a little easier to give an example first. Think about what you know about the integers. Sometimes I will use the term element. This essentially means a number that is an integer.
(2.1) Think about addition in the integers. When adding them, you can always find 2 numbers that when added will give you zero. For example, 
. I can name any number 
and you will always say that adding 
will give us zero. In math, we say that all the elements under addition have inverses. We call 0 the identity of the integers under addition because if we add 0 to any number, then we still get that number. For instance: 
. More over, adding any two integers together will produce another integer, like 
and 
is still an integer. We say that means that the elements are closed under addition. Finally, elements commute so the order that you add the integers won’t change your answer, ie 
or 
.
So the punchline is this: because the integers are closed under addition, have an identity element, each element has an inverse that gets us back to the identity, and the elements commute, we say that the integers are an abelian group under addition. Any “group of numbers” that satisfy these properties are also called an abelian group. If the elements don’t commute, then we just call it a group.
Notice that I said the integers were a group under addition. The integers are NOT a group under multiplication. Why? The elements are closed, and the identity element is 1 (because 
). However, the integers do not have multiplicative inverses (what integer can you multiply again 3 to get 1? None.). The rational numbers are a group under multiplication though.
(2.2.) Now let’s go back to thinking about multiplication in the integers. We know they don’t form a group, but that’s okay right now. The only thing I want to bring your attention to is that the integers are associative. This means that the order of multiplication does not matter so long as the sequence of numbers stays the same, for example:
.
Also, remember that the multiplicative identity of the integers is 1. That means we can multiply against any integer and still get that integer. So under multiplication, we have associativity and an identity element.
(2.3.) Finally, we can distribute the integers. That means we can do something like
or 
.
Now, if you are with me so far, we can go ahead and define what a ring is.
A ring is any set (like the integers) that fulfills these properties: (1) the set is an abelian group under addition (2) multiplication is associative and has an identity element (3) you can distribute the elements. (See ** Below for An Additional Detail)
3. Integral Domains and UFDs
Okay, so if you are still with me, we are getting to the point soon, I promise! Now that you understand what a ring is, you have to realize we can add more properties on top of the three I told you about in the last section and get more “specialized” rings.
So, there are two you need to know about. The first is an Integral Domain. An integral domain is a ring that has no zero divisors. The means there are no elements besides zero that when you multiply the elements together you get zero. The integers are an integral domain, because the only time you get zero under multiplication is when you multiply an element by zero (
). (* Now, I would suggest reading below under the “additional notes” section to see a simple example of what would Not be an integral domain if this doesn’t make total sense to you) That’s it. That’s all an integral domain is.
Now let’s move on to Unique Factorization Domains, or UFDs for short. You probably can already guess as to what makes a UFD a UFD, but let’s go ahead and formalize it. A UFD is an integral domain where every element can be factorized uniquely into irreducible elements. Notice I said irreducible and not prime. So what’s the difference between irreducible and prime? Well, in a UFD, there is no difference. But I’m getting ahead of myself, give me another minute and then I’ll explain what that means. An example of a UFD is the integers, because (as we discussed already) the integers have unique prime factorizations.
4. Polynomials and Adjoining Elements
The last thing you really have to understand is polynomials. Before I go any farther, I want to introduce some notation. I have been talking about the integers alot, and I’m going to denote them as 
from now on.
Now, you know what a polynomial is. Formally, it’s a function of the form 
… 
where the 
are integer coefficients (for our purposes they are integer coefficients). Examples are:
or
.
So. Nothing new here. When we are talking about such polynomials with integer coefficients, we denote it with the notation ![\mathbb{Z}[x]](http://s.wordpress.com/latex.php?latex=%5Cmathbb%7BZ%7D%5Bx%5D&bg=ffffff&fg=000000&s=0 "\mathbb{Z}[x]")
. (Note is a ring. Can you prove it? It’s also an integral domain.)
Now, I’m not going to formally tell you what adjoin means but I will show you how we will adjoin elements. Say I wanted to adjoin 2 to . Then I could just put 2 in place of 
in to get ![\mathbb{Z}[2]](http://s.wordpress.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B2%5D&bg=ffffff&fg=000000&s=0 "\mathbb{Z}[2]")
. I can actually pick whatever number I want. For instance, I might choose to look at ![\mathbb{Z}[\sqrt(2)]](http://s.wordpress.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B%5Csqrt%282%29%5D&bg=ffffff&fg=000000&s=0 "\mathbb{Z}[\sqrt(2)]")
or ![\mathbb{Z}[\sqrt{-5}]](http://s.wordpress.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-5%7D%5D&bg=ffffff&fg=000000&s=0 "\mathbb{Z}[\sqrt{-5}]")
. All it means, for our purposes, is that where ever you see an in a polynomial, you replace it with that element. So suppose you have
and we decide to look at . Then this is just:
For our purposes, this is how to think about adjoining elements. Notice that we no longer have the element in any of our polynomials – it is always replaced with whatever number we adjoin.
5. Primes and Irreducibles in Modern Algebra
Whew. If you made it this far, then you are finally getting to my point. Woo hoo!
Irreducible sounds like a pretty obvious word: you can’t reduce the element any farther. You may be wondering why we don’t just use the word prime instead of irreducible, and this finally brings me to my point.
In , irreducibles and primes are the same thing. Obviously you can’t reduce 5 any farther – it’s prime! But here’s the trippy thing: in an integral domain, irreducibles and primes are not the same thing.
So, let’s go back to . First of all, it’s an integral domain. Now think about the number 9. From earlier, we know 
. But in ![\mathbb{Z}[\sqrt{-5}], 9 = (2 + \sqrt{-5})(2 - \sqrt{-5}) = 4 -2\sqrt{-5} + 2\sqrt{-5} - (-5)](http://s.wordpress.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-5%7D%5D%2C%209%20%3D%20%282%20%2B%20%5Csqrt%7B-5%7D%29%282%20-%20%5Csqrt%7B-5%7D%29%20%3D%204%20-2%5Csqrt%7B-5%7D%20%2B%202%5Csqrt%7B-5%7D%20-%20%28-5%29&bg=ffffff&fg=000000&s=0 "\mathbb{Z}[\sqrt{-5}], 9 = (2 + \sqrt{-5})(2 - \sqrt{-5}) = 4 -2\sqrt{-5} + 2\sqrt{-5} - (-5)")
. But 
does not divide 
and vice versa, thus the prime factorization of 9 does not exist and 3 is not a prime in this ring! Thus an irreducible number does not mean the number is prime in an integral domain. However, a prime number is still consider irreducible in an integral domain. Crazy!
Now, if you are freaking out that all hope is lost for making any sense of primes and irreducibles ever, fear not! This sort of behavior is only possible in an integral domain. This can’t happen in a UFD, which is what the integers are. Any other type of rings we can classify beyond a UFD, primes are irreducibles and irreducibles are primes. Thus, for most people, this is never something you learn about. However, in graduate school, suddenly you have to worry if you can call the element prime or irreducible in certain scenarios.
So that’s it. Yes, it took me a long time to get to this place. I hope, if you are still reading, that you have some deeper appreciation for pure mathematics and some of the weird things that can happen when you go out of the realm of your comfort zone. If you are interested in finding out more about ring theory (what I presented is very very very basic), then please see the links below!
Additional Notes
* What doesn’t qualify as an integral domain? The simplest example of this requires that you understand the concept of modular arithmetic. Think back to high school when you did division with remainders. Consider the following calculations (where R is remainder…I think that’s how we did it in high school):
Obviously, every time you run into a number that has a multiple of 3 in its prime factorization you have remainder 0. Otherwise, the only other possible remainders you can have are 1 or 2. In mathematics, instead of saying “remainder 1″ we say 
. So 
means that is the remainder between 0 and 2 after dividing by 3. Such sets are groups under modular arithmetic. So every number that we divide by 3 can be reduced to 
or 
(sorry if I’m beating a dead horse here).
Now what does this have to do with integral domains? When you mod out by a number that isn’t a prime, the set does Not form an integral domain. Let’s looking at modular arithmetic using 3 again. Suppose we have , where is not 0 (So could be 1 or 2, just like above). Then multiply 3 against produces the following result: 
. This makes sense because 3 is essentially zero when we are working with modular arithmetic, so this example IS an integral domain.
Now, let’s look at modular arithmetic using 4. Suppose we have 
. Then 
. Then 2 is considering a zero divisor, because remember 
in modular arithmetic. What I’m saying is, modular arithmetic with 4 is NOT an integral domain.
Can you generalize for what numbers we are guaranteed to have integral domains?
** Amended thanks to Aris. See also: Beyonce. If you liked it then you shoulda put a ring on it.
Additional Reading
If you want to learn more about this, please see the following wikipedia pages:
Modern Algebra
Ring Theory
Integral Domains
UFDs
Group Theory
Number Theory
Fundamental Theorem of Arithmetic
Modular Arithmetic
Chinese Remainder Theorem
16.03.10 / grad school, hobbies that aren't math / Author: JHill / Comments: (0)
It’s Spring Break. Despite having gobs of work to do, this is the first time I actually feel like I’ve taking a break from school. Working during the day, and playing in the evening.
Of course I’m in Chicago. It’s the longest period of time I’ve been in the city since I moved, which is really kind’ve tragic. Regardless, it’s wonderful to be here.
I had one of those moments today that reminded me what I love so much about cities, and particularly this city.
I went to a bookstore to work, and a man sat next to me with a pile of art books. Now, you should know that I have a tendency to have long fascinating conversations with complete strangers, so when I am working it is my policy to not talk to someone until I am done working (lest I derail myself from being productive). He began sketching patrons in the store, and he was really quite good.
As I packed up, I was sure to mention to him that he was a great artist. He asked what I’d been working on, which of course was math. It turns out he actually does unemployment projections for the Department of Labor. This, of course, is an endlessly fascinating topic to me. He had a masters in economics, and did art as a side passion. It was such a fantastic conversation, which left me feeling revived as I walked home.
People fascinate me. I took a vacation to New York by myself with the intent of talking to as many people as I could. I met so many great people on subways, in restaurants and on the street. My impression of New York is that people are very lonely and are dying for someone to ask them about their life stories. This is a great position for me to be in, because life stories open my eyes to the world. This kind of interaction can really happen anywhere though, so I shouldn’t generalize. It just tends to be easier to find it in a big city.
Seeing as its my first full day home, it made it really great to be back.
