Dreams of Fields

Math is hard...............Barbie
Doh!.............................Ken.

Math is one of many things that seems harder as I get older. I am studying a rather easy math book, but lately am having a bit of trouble solving the problems.

A field is a set of number like things that can be added, subtracted, multiplied and divided, with the result always being in the field [UPDATE: except division by zero is not allowed!]. The integers, for example, are not a field, since their ratios aren't always integers, but the rational numbers (positive and negative ratios of numbers, aka fractions) are. Additional infinite fields include the real and complex numbers. There are also finite fields, the simplest one consisting of just 0 and 1, with 1+1 defined to be = 0. (Arithmetic modulo two, or Z2). Doing our arithmetic mod 3, mod 5, or mod 7 also works, but not mod 4 (or 6, or 8, or 9...). The problem is that you can't have a field when you have zero divisors (e.g., 2 X 2 mod 4 = 0).

Galois, besides inventing group theory and getting himself killed in a duel at age 20, invented a way to build new fields out of old, a method named now after Kronecker - Mathematicians have a habit of naming things after people other than the discoverers. The implications of that Kronecker construction are what is giving me trouble.

It's a very important idea, the key, in some ways, to some of the most fundamental ideas of algebra. We recall from our high school algebra that we were always trying to find X, the unknown, about which we typically knew only that it satisfied some sort of polynomial equation such as: X^2 + 2 X + 1 = 0. What sorts of polynomial equations could be solved, and how, preoccupied algebraists until Galois (and others) sorted things out early in the nineteenth century. The mathematics they invented turned out to have implications far beyond the original problems.

And that's what I'm trying to understand a little bit of.