TITLE: In plain language, what's the significance of a field?
QUESTION [47 upvotes]: I just started Linear Algebra. Yesterday, I read about the ten properties of fields. As far as I can tell a field is a mathematical system that we can use to do common arithmetic. Is that correct?

REPLY [26 votes]: Yes, at its most basic level, a field is a generalization of the rational numbers. In a field, you can do addition, subtraction, multiplication and division as you could in $\Bbb Q$.
At a deeper level, fields have geometric significance. If you've ever studied a little geometry,  then you'd know that there are at least two famous ways to approach geometry: with axioms akin to Euclid's axioms (the synthetic approach) and another way by using vector spaces and equations (the linear algebra approach).
We know that $\Bbb R\times \Bbb R$ (an $\Bbb R$ vector space) can be interpreted as a model of Euclidean geometry, and how its 1-d subspaces represent lines, it's elements represent points etc, and that it satisfies the synthetic axioms of Euclidean geometry. 
But what about the other direction? Why can't we start with synthetic axioms and get vector spaces? Well, that's the thing: you can (if you have enough axioms.)
It turns out that if you adopt Hilbert's axioms groups $I-IV$ for plane geometry, then you can systematically build a field $F$, such that $F\times F$ models that plane exactly when the plane satisfies Pappus's theorem.
Another way to ensure the existence of the field is to adopt Hilbert's continuity axiom $V$ called "Archimedes's axiom." It's known that this axiom, in the presence of the others, implies Pappus's theorem, and the resulting field will be an Archimedian ordered field.
You can, of course, do higher dimensional geometry and get vector spaces $F^n$ and so on, as long as you have something like Pappus's theorem or Archimedes's axiom in your axioms.
If you asked me for a rough description of how the axioms of fields translate into geometric ideas for vector spaces, then this is how I would start. Since $F$ is an additive group, $F^n$ is also an additive group, and you can translate any point to another point using addition of vectors. For multiplication, you can use it to scale any vector to another vector in the same 1-dimensional subspace.
Now, this is just the first hint at the geometric nature of fields. Galois theory and then algebraic geometry really take the connection to more extreme altitudes!

REPLY [11 votes]: Certain items keep showing up in mathematics. When they show up often enough, we give them names. 
"Field" is an example of this. You know about the real numbers: they're things that you can add and subtract, multiply and divide. Addition is commutative. So is multiplication. And multiplication distributes over addition.
It turns out that there are other sets that have all these properties, like the Complex Numbers. 
There are some other sets that are kind of interesting, like the set {O, I} (those are the letters "O" and "I"), on which I can define addition by the rules $O + O = O, I + I = O, O +  I = I + O = I$. That might look silly, but you can check that there's an identity (O), negation (the "negation" of each element is itself!), and addition is commutative. I can then define multiplication: $O*O = O*I = I*O = O; I*I = I$. Together, these have all the properties I mentioned above, and this is called "the finite field with two elements".
Because these properties let you manipulate objects using the rules of algebra you learned in high-school, they're pretty convenient to work with. And if you ONLY use those rules, then whatever you do will work not just for the reals, but for the complexes and various finite fields, and lots of others. So proving things about general fields gives you more power than proving things just about the real numbers. 
(Fields generally DON'T have to have a notion of "less than" and "greater than", so when you're working with them, you have to forget that part of high-school. :)  )