Though the connection may not at first be apparent, this is part of my promised (threatened?) attempt to put the fundamentals of Galois theory in terms suitable for readers of this blog. It will be a slow build, because there are a lot of a pieces to put into play.
Today, a thought experiment. Imagine you have made contact with another form of intelligent life. Communication is still at a primitive stage, but you’ve devised a way of sending each other signals, and you and the alien are in the process of building up your shared vocabulary in this new language. (I’m imagining some sort of IM window, your imagination may vary.)
Well you’ve heard that the universal language is mathematics, and you want to establish a shared vocabulary for basic math. With some effort, you establish an agreement on the concepts of “addition” and “multiplication” (think about how you might do this, how you might distinguish these two operations from one another). You figure out what name they have for what you call “zero” and “one” easily enough. (For example, you could ask what number plus itself equals itself to nail down zero, then ask what number multiplied by itself equals itself, other than zero, to nail down one — think about it.) Once you have zero and one, addition and multiplication, you can get 2, 3, 4, etc., then the negative integers, and then fractions.
It would take some time, but suppose you eventually get sufficient communication to have shared language for the real number line (maybe you explain Dedekind cuts, whatever, I don’t care). (Actually this isn’t essential, and it’s just as interesting to suppose you don’t establish shared vocabulary for the real numbers; we’ll explore that elsewhen.)
So now you’re feeling ambitious, and you want to know how the alien talks about imaginary numbers. What does the alien call your ? You assume (reasonably) that such a developed race would also have some corresponding concept, so you ask for a number which multiplies by itself to give negative one, and the alien says “blarg: blarg times blarg plus one is zero”. Victory!
But then doubt sets in. Are you really sure his blarg is your ? After all, too. Maybe blarg is negative ? How would you know? Think about it as long as you like, but the answer is, you wouldn’t. There are no questions you could ask that would say for sure whether blarg was or .
(You might try to say something about “the one on the upper half of the complex numbers”, but that’s no good. You have no reason to believe that they visualize complex numbers anything like how you do, and anyway that distinction is happening only in your mind, not in the math. It’s no more constructive than defining “three” as “the number that looks like half an eight”. That’s not math, not even arithmetic. It’s trivia about our way of writing numbers.)
We could rephrase this whole thing without aliens (but why would you ever prefer not to include aliens?). Suppose that I had misunderstood my teacher the day she defined the complex plane; suppose I had thought that was one unit below the origin, the opposite of the convention you’re probably used to. What would happen when I try to talk math with the people like you who learned it the usual way? Nothing interesting! You and I believe all the same statements about numbers! We both think and we both think . If we visualize these facts geometrically, then the picture in my head doesn’t match the picture in yours (it’s upside down). As long as we stick to the numbers and equations, as long as nobody explicitly mentions the pictures we are thinking about, we’ll be in perfect agreement about complex numbers.
You may have learned in high school that, if you have a polynomial with real coefficients and is a root, then so is . Now we see the reason that underlies this truth: no algebraic statement in terms of real numbers can distinguish from one another. The point in your mind I call , I call , and vice versa.
In fancier talk: the complex numbers have a symmetry, usually called complex conjugation, which preserves all the real numbers and which preserves any facts and relationships which can be expressed in terms of basic algebra. The numbers and are interchangeable because they have to be, because they are bound by the symmetry. Symmetries are magical things.
As we shall see, symmetries are powerful tools for understanding many kinds of situations, and the language of mathematics is the right language for getting at symmetries.
But there is more to the story. We’ll talk to the aliens a little more next time.