Fields

This post is actually for my brother, a very smart non-mathematician who regularly asks me insightful questions about mathematics.  Over Thanksgiving, he was probing me about a piece of abstract algebra, the existence of different sorts of fields. I’m going to rehash much of the discussion we had, then add a few thoughts I didn’t get to discuss in person.  (This was also written somewhat hastily, so if there are errors or things that don’t make sense, please let me know!)

What are fields?

A field is a collection of numbers together with operations called “plus” and “times” that behave in the way operations with these names “should” behave; in particular, there is a number called 0 which, if added to any other number, leaves it unchanged (an additive identity); similarly there is a multiplicative identity called 1, and we have the usual rules like the ability to subtract and to divide by numbers other than 0, associativity and commutativity of multiplication and division, and the distributive property of multiplication over addition.

As my brother noted, the historical development of mathematics has involved a lot of expansion in what we think of as a “number” — for geometric reasons, the set of rational numbers was expanded to the larger set of real numbers, which includes irrationals like $\sqrt{2}$ and $\pi$; then for algebraic reasons, this was expanded again to arrive at the complex numbers.  All these collections of numbers are fields.  (There are more esoteric extensions of this number system, like the quaternions (my brother was not impressed with this name), which are no longer a field.  Despite a great deal of geometric meaningfulness they have never been the sort of thing that is common knowledge for non-mathematicians, though perhaps they will feature in the 23rd century high school mathematics curriculum.)

Examples of fields

There are also other fields, as well; for example, the set $\mathbf{F}_2 = \{0, 1\}$ with the rule $1 + 1 = 0$ (and other rules of arithmetic as usual) forms a field, with only two elements.  (In fact, my brother observed the subtle point that under the definition above, there is a field with one element equal to both 0 and 1; often the formal definition of a field requires $0 \neq 1$ to rule this case out for convenience in stating certain theorems.) And just like we can expand the real numbers by adding new algebraic numbers (like i, a root of the equation $x^2 + 1 = 0$), we can expand these other fields.  For example, it’s easy to check that in $\mathbf{F}_2$, there is no solution of the equation $x^2 + x + 1 = 0$.  (There are only two things to check: whether 0 is a solution and whether 1 is a solution.)  So, we can add a new solution, called say j, and add it to $\mathbf{F}_2$, then add all the other numbers we need to make the resulting object a field.  As it turns out, the resulting field $\mathbf{F}_4 = \{0, 1, j, j + 1\}$ has exactly four elements, which obey rules like $1 \div j = j + 1$.  This technique of extending a field by adding the solution of a polynomial equation is called algebraic extension; sometimes, one finds a field (like the complex numbers C) such that it already contains all of the solutions of the polynomial equations one can write down using that field; in this case we say the field is algebraically closed.  Every field is contained in some algebraically closed field, in particular the algebraic closure that you get by simultaneously adding all the roots of all the polynomial equations.

Field characteristic

One interesting fact about the finite fields (like $\mathbf{F}_2$) is that their algebraic closures actually end up infinite — you can’t pack all the solutions of all the polynomial equations into just a finite number of elements.  So one question is whether you just end up with the complex numbers C as the algebraic closure of everything.  It’s not hard to see that this is not the case: no matter how you extend $\mathbf{F}_2$ by adding elements, you will still always leave intact the fact that 1 + 1 = 0.  But of course in the complex numbers $2 \neq 0$, so the fields cannot be the same.  This leads us to a natural property of fields, called the characteristic: suppose you have a field F and you start computing the sequence 1, 1 + 1, 1 + 1 + 1, 1 + 1 + 1 + 1, ….  When does this sequence wrap around and get back to 0?

In the fields we’re used to (the rational, real and complex numbers), of course, the answer is “never”.  But as we’ve seen, in $\mathbf{F}_2$ and $\mathbf{F}_4$, we get back to where we started in just two steps.  One can also make fields with other characteristics; for example, the elements 0, 1, 2 together with the rules 1 + 2 = 0, 2 + 2 = 1, and $2 \times 2 = 1$ (and all other arithmetic as usual) form a field; this is modular arithmetic (or “clock arithmetic”) modulo 3, and the characteristic of this field is 3 (since 1 + 1 + 1 = 1 + 2 = 0).  In fact, for any prime p, there exist fields of characteristic p.  One can also show that only prime characteristics are possible.  For example, if you give me a field allegedly of characteristic 6 (so containing elements 0, 1, 2, 3, 4 and 5 such that 1 + 5 = 0) we have that $2 \times 3 = 6 = 0$; dividing both sides by 2, we get 3 = 0, so really we had to have wrapped around sooner than the 6 that we claimed.

Some fields of characteristic 0

Those fields for which we never get back to 0 by repeatedly adding 1s (like the rational numbers) are said to have characteristic 0.  As it turns out, there are lots more of them than just the most common ones; for example, if we take the rational numbers and add to them just $\sqrt{2}$, the smallest field containing the result is exactly the set of numbers of the form $a + b \sqrt{2}$ where a and b are rational numbers.  (This is not totally clear: you have to check that, for example, $1 \div (1 + \sqrt{2})$ can be written in this form.)  This field lives somewhere between the rational and real numbers.  There are also fields that live between the rational and complex numbers neither contain nor are contained in the real numbers; for example, the set of numbers of the form a + b i (where i is the square root of -1) with a and b rational forms a field.  (Note: this field is not the algebraic closure of the rationals — you also need to add in things like $\sqrt{2}$.  But the algebraic closure of the rationals is still smaller than the complex numbers — you don’t need to include transcendental real numbers like $\pi$.)

Less obvious fields of characteristic 0 are ordered fields that contain the real numbers.  (The complex numbers can’t be ordered, that is, given a sense of “positive” and “negative” that works well with multiplication and addition: no matter whether i should be positive or negative, $i^2$ should be positive; but of course $i^2 = -1 < 0$.)

A little motivation is in order: back at the dawn of calculus, Newton and Leibniz developed their techniques using the idea of an infinitesimal quantity, something positive but smaller than all the real numbers.  Their infinitesimals didn’t behave like nice elements of a field (the product of two infinitesimals had an uncanny habit of getting set equal to 0 at convenient moments), and later mathematicians (especially the French mathematician Cauchy, whose politics my brother would have despised had they been contemporaries) ended up circumventing their use by introducing the idea of limits (with which present-day mathematicians regularly torture their calculus students).  But, as it turns out, there are perfectly good ways to extend the real numbers by adding infinitesimals.  Begin by adding a single number $\varepsilon$ to the reals, and declare that $0 < \varepsilon < r$ for every positive real number r (but otherwise don’t impose any new conditions).  Now we have lots and lots of new numbers to add to our field, like $\dfrac{\varepsilon^2 + \varepsilon}{\varepsilon^4 - 6\varepsilon + 1}$.  (In fact, it happens that all of these elements take the form of a fraction whose top and bottom are polynomials in $\varepsilon$.)  Remarkably, we can figure out for each pair of these elements exactly how they compare to each other.  For example, the number I just mentioned is also infinitesimal, but is a tiny bit larger than $\varepsilon$.  (Why?  Well, the numerator is $\varepsilon$ times something just a bit larger than 1, while the denominator is something just a bit smaller than 1.  So when you divide, you get something just a bit larger than 1 multiplied by $\varepsilon$.)  But there are also infinite quantities, like $\dfrac{1}{\varepsilon}$, which is larger than every real number.  And, in fact, there are infinite hierarchies of infinite numbers: $\left(\frac{1}{\varepsilon}\right)^2$ is not only larger than $\dfrac{1}{\varepsilon}$, it’s larger by a factor that is itself infinite!  And, in fact, it turns out that all of the sort of sketchy stuff that Leibniz and Newton were doing with infinitesimals can be put onto a completely rigorous footing in this setting, which allows one to redo all of calculus without needing to ever use the word “limit” at all.  This set (sometimes called the hyperreals) is among my favorite things in all of mathematics.