Fields! Why are they interesting and important?

So, now that we’ve gone over the prerequisites, what are fields in a mathematical context? Well, if you’ve just come from the Multiplicative Inverse axiom, this will look familiar. The proper definition of a field is a commutative ring R with identity “1” not equal to “0” that satisfies the following axiom: for each a not equal to “0” in R, the equation ax=”1” has a solution in R. Note that “1” refers to the multiplicative identity and “0” refers to the additive identity. The definition is basically all encompassing, taking each of the previous axioms and combining them together. Now, you will hopefully notice that the definition for an Integral Domain actually isn’t listed here. This is because a field doesn’t need that definition to exist. So, why did I say that a field encompasses all axioms, including the integral domain? Well, it turns out that the integral domain axiom actually does hold in fields and can actually be proven by the other axioms, so we don’t include it here even though it does exist in a field.

I’ve been dancing around this topic for most of the other pages, but I now get to talk about it in the open. The real numbers, the actually numbers most of us use in our daily lives, are a field! This is why I could, if I wanted, use the real numbers as an example for every axiom, and it’s because they are a field which is defined to follow all of these axioms. Other number systems that follow this are some modular arithmetic applications such as Z_6 and, interestingly enough, the complex numbers! On the next page, I will help you take a look at the complex numbers and some interesting things about them that will hopefully increase your understanding of them.

So, why are fields important to us? Well, the easiest way is to say that they exhibit properties that are extremely useful to us. There is a reason we use the real numbers instead of some other strange number system. It’s because it exhibits all the 12 axioms we have gone over, and thus it makes it easy to work with. While there can be uses for things such as Rings, for us average people, anything that is a field doesn’t really require us to figure out “what works”. We can just go about expecting things to behave a certain way, which is what we do normally when we are working with the real numbers.