In plain language, what's the significance of a field?

Solution 1:

More or less, yes. You can do arithmetic already with natural numbers (in $\Bbb N$), but you have to be careful with some operations:

The equation $x+a=b$ does not necessarily have a solution.

So one would prefer working in a ring, like the integers ($\Bbb Z$). The preceding equation has always one solution, $x=b-a$.

But then, the equation $ax=b$ does not have always a solution, in a ring.

So one would prefer working in a field, like rational numbers ($\Bbb Q$). The preceding equation has always one solution, $x=\frac ba$, if $a\neq0$.

So, working in a field is much more friendly. However, there are strange fields too, like finite fields. Or quaternions, for which commutativity does not hold anymore ($ab\neq ba$ usually): they form what is called a skew-field.

Just to develop this way of building larger tools a bit more:

But even then some equations have no solutions, like $x^2-2=0$ or $x^2+1=0$. So you may build larger fields: real algebraic numbers, complex algebraic numbers, real numbers ($\Bbb R$), and finally complex numbers ($\Bbb C$).

The field of real numbers has a valuable property, very important in analysis: every Cauchy sequence has a limit. That is, $\Bbb R$ is complete. It's for example easy to prove that the sequence of rationals $u_n=1+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}$ is a Cauchy sequence, but does not converge to a rational.

The field of complex numbers is also complete, and have the extra feature that every nonconstant polynomial can be factored in factors of degree $1$. It's said to be an algebraically closed field. However, you lose something too, since, unlike the reals, it's not an ordered field: there are total orders on complex numbers, but none is compatible with the arithmetic operations.

As you can guess, while building larger and larger sets, you are more and more generalizing what you mean by a number.

Other genralizations of numbers that form a field include hyperreal numbers and surreal numbers.

The story does not end here, and many mathematical objects that form a field do not resemble numbers very much, apart from having the basic arithmetic rules of a field.

Given any integral domain, you can build its field of fractions:

• For instance, the ring of polynomials over a field $\Bbb K$ leads to the field of rational functions over $\Bbb K$. Finite extensions of such fields yield algebraic function fields.
• Formal power series over a field $\Bbb K$ lead to the field of formal Laurent series. An algebraic closure of the field of Laurent series is given by the field of Puiseux series.
• Holomorphic functions defined on a complex domain, lead to the field of meromorphic functions.
• In algebraic geometry, functions fields are defined on an algebraic variety, again as a field of fractions.

Another example of a field not related to the notion of number: Hardy fields are fields of equivalence classes of functions.

Solution 2:

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!

Solution 3:

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. :) )

Solution 4:

Have you discovered yet that finite fields also satisfy the field axioms? Then the arithmetic is rather uncommon. For example $F_p$ is the field of integers modulo $p$, a prime. In $F_7$ we have things like $5\times 4=-1$. But it still has many nice properties.

However, there is a theorem, basically easy, but tedious, to prove that if you also add an order axiom (we have things like $x>0$ that behaves the way you would expect) and a "completeness" axiom (that every let has a least upper bound) then the field must be essentially the ordinary real numbers (ie just the reals relabeled).

Note, for example, that any sensible order must have the property that if you start with 0 and keep adding 1, you get bigger and bigger numbers. That must be false in a finite field.

Solution 5:

Here's the plainest way I can think of:

A field is a place where you can do division and multiplication commutes.