People mean different things by “number” at different times. Sometimes “numbers” means integers, or counting numbers, or real numbers, or fractions, complex numbers, or maybe something else entirely. There are a wide variety of number systems that we might use in various contexts. Today I will describe fields, a certain kind of algebraic structure which expresses exactly what I mean by “number system” for the purposes of Galois theory.
For those of you still reeling from the definition of a group, don’t worry. A field is a much easier thing to get your head around. It’s just a collection of things (I’ll tend to call the things in a field “numbers”) where we have the four basic operations (addition, subtraction, multiplication, and division by anything but zero) and the operations behave the way you’re used to.
What do I mean by “the way you’re used to”? I mean the following.
- addition is commutative ()
- addition is associative ()
- 0 is the additive identity ()
- subtraction is the “opposite of addition” in the sense that
- multiplication is commutative ()
- multiplication is associative ()
- 1 is the multiplicative identity ()
- division is the “opposite of multiplication” in the sense that , assuming .
- multiplication distributes over addition ()
To use the lingo of group theory, we are saying that the elements a field form a commutative group under addition, that the elements (except 0) form a commutative group under multiplication, and that the two operations are related by the distributive law. But if that’s confusing, forget what I just said.
Digression. When I was a child, I wondered why mathematicians could invent imaginary numbers so that we could take square roots of negative numbers, but no one ever tried to similarly finesse division by zero by making up a new number to play the role of . Why not? Why was division by zero so much harder than square roots of negative numbers. Notice by the way that, if any is any number whatsoever in any conceivable field, then the distributive law . That is, the common notion that anything times zero is zero is fully universal. It’s not just a statement about real numbers, it’s a statement about what it means to be a number. That’s the difference. No real number squared is negative, but that’s an obstacle specific to the real number line. No number of any kind, in any context we would recognize, times zero is one.
You already know about several important fields. Here are some examples
- , the field of all rational numbers
- , the field of all rational numbers
- , the field of all complex numbers
- , the field of integers modulo 11 (convince yourself that division really works here!)
- , the field of all complex numbers ( integers) for which both the real and imaginary part happen to be rational numbers (again, it is not totally obvious that division works out right; the key is rationalizing the denominator, like Mrs. Gunderson taught you in school)
There are also some important nonexamples. In some other contexts, these may be called number systems, but they are not fields, the kind we care about now.
- , the integers
- , the integers taken modulo 12 (why is this different than modulo 11?)
A particularly interesting situation occurs when one field contains another, as the complex numbers contain the real numbers. Note that every complex number has an expression in the form , where are real. Whenever there is a finite list of elements and every element in the larger field has a unique expression in the form , we say that the larger field is a finite extension, and is the degree of the extension. So the complex numbers are a degree-2 extension of the real numbers (and the natural basis is ).
The usual way to manufacture finite extension fields is to begin with a field and some polynomial that does not have solutions in the field, and then enlarge the field just enough so that the polynomial can be solved. This is how arises, by beginning with and adding solutions to . This will always produce a finite extension field. In fact, it turns out that in some sense this is the only way to get finite extensions. That is, if is a finite extension of , then there is some polynomial over such that is obtained by adding a root of to .
Finite extensions of have a special role in mathematics; they are called (algebraic) number fields.
(Most of what I just said is intuitively plausible, though some of it may be surprising. Almost none of it is obvious, but proofs can be found in any introductory algebraic number theory text for those who want.)
I have said that Galois theory is about the symmetries of number systems. If is a field and is some field containing , then we will want to study the symmetries of over ?
So what, exactly, is the right thing to mean by a symmetry of a number system? In the case of over , it was somehow clear that the right things to consider were complex conjugation and the trivial symmetry. But more generally, what is a symmetry of one number system relative to another? And what can we learn by understanding them?
I’ll tell you next time, Gadget.