The Big Idea

So, in the introduction, I avoided giving any reference to what my project is actually called, or in general what it’s about.  I’ll change that here.  Kind of.

Broadly speaking, I’ll be researching a topic in ‘algebraic number theory’.  I expect that will mean very little to most people reading this at the moment, so let me explain.  Algebraic number theory (ANT) is a branch of number theory (shocking!).  Number theory itself is the in-depth study of integers (whole numbers) and what interesting properties they have.  A particular collection of integers that is of interest, for example, are prime numbers (2, 3, 5, 7, 11…); numbers that are divisible by themselves and 1.

Some of number theory is very easy to understand. For a famous example, Pythagoras’ Theorem can be seen to boil down to which whole numbers work for the equation . However, this doesn’t work when we look at ,  or indeed any power greater than 2; something known as ‘Fermat’s Last Theorem’. A less-well known number theory statement is Goldbach’s conjecture: Every even number greater than 2 is the sum of two primes (4=2+2, 6=3+3, 8=5+3…). It’s a really simple statement to understand. However, how do you prove these things? Do we know that, say, 79494819480 is the sum of two prime numbers?  It’s not exactly easy to show…

This is what I love about number theory. Even though some of it is very simple, what’s going on ‘behind the scenes’ if you will is very complicated, and even the easiest and most obvious of things requires lots of difficult maths to prove, so that we can be sure it’s correct!

So, that’s number theory in a nutshell. But what about ANT? Well, that’s a bit more complicated to explain. Earlier, I said that number theory is about whole numbers? ANT decides that’s a bit boring, so decides to study things called algebraic numbers. I’ll explain what they are properly later. However, for the time-being, it’s worth saying that all whole numbers are algebraic numbers, but not all algebraic numbers are whole numbers. So, for example, 2 is a whole number and an algebraic one, but is an algebraic number, but not a whole one (again, I’ll explain why later). 

Basically, ANT broadens the numbers we are interested in. Why? By doing this, we can get answers to number theory problems which we may not have gotten otherwise. It’s sorta like you’re driving along a road and then all of a sudden large pile of bricks is put in your way. You could either try and move all the bricks so you can take the obvious route, or you can just leave them alone, and take a detour that adds 10 minutes extra driving time. It may seem weird at first, but it makes sense when you look at the small details.

And that’s algebraic number theory. As I’ve said, my project is an algebraic number theory one. However, my project isn’t about trying to get an answer in number theory, it’s simply ANT for the sake of ANT. It’s not one about trying to get past the ton of bricks, it’s one about enjoying the detour you have to take. What exactly I’m doing, I’ll be explaining much later.

Hopefully that’s given a ‘big idea’ to what it is I’m doing. I’ll soon be going into the more ‘nitty gritty’ stuff…yay!