Cam's Blog

November 16, 2009

Symmetries of numbers part 2

Filed under: Elementary, Math — cfranc @ 12:18 am

This is a continuation of an earlier post, which is aimed at explaining what symmetries of numbers are to an elementary audience. The only prerequisite to understanding is a willingness to learn. I’ll begin by summarising the material discussed last time.

Our discussion began by trying to formalize everyone’s natural and intuitive understanding of geometric symmetry. To do this we considered geometric objects living inside of another space, like a plane or three-space. To be concrete, let’s say we’re working in the plane. Then the symmetries of the plane are just the distance preserving maps from the plane to itself. That is, a symmetry is a rule which transforms every point in the plane to another point. This rule must satisfy the property that if two points are at distance d from one another before being transformed, then they must also lie at distance d from one another after the transformation takes place. For instance, rotation of the plane about a point is distance preserving. Now, given an object living in the plane, say a square, then a symmetry of the square is nothing but a symmetry of the plane which preserves the square. For instance, rotating the plane by 90 degrees about the center of the square gives a symmetry of the square.

After this we described the arithmetic object which plays the role of the plane in what follows. We recalled the notion of a complex number (remember that the set of all complex numbers is denoted \mathbb{C}), and defined an algebraic number to be a complex number which is a root of a polynomial with rational numbers as coefficients. For instance \sqrt{2} is a root of the polynomial X^2 - 2, which has rational coefficients, so that \sqrt{2} is an example of an algebraic number. We denoted the set of all algebraic numbers \mathbb{A}, although one usually writes this set as \overline{\mathbb{Q}} (read it as Q-bar). That brings us to the end of the previous post.

Our analogy with the plane above has hopefully made it clear that our next goal is to define symmetries as certain transformations f \colon \mathbb{A} \to \mathbb{A}. Just as geometric symmetries preserve geometric properties, ours should preserve arithmetic properties. So before we can touch the question of symmetry, we must first examine what arithmetic structure is possessed by the set of algebraic numbers. Unfortunately I don’t know of any way to prove some of these facts in a completely elementary way, so instead I’ll state some facts and try to illustrate by example. The important points are that algebraic numbers can be added, subtracted, multiplied and divided (as long as the denominator is nonzero). Now, algebraic numbers are complex numbers, and all complex numbers can be added, etc. However, it’s not clear that the sum, product, etc, is not just complex, but is in fact itself algebraic. In other words, if you take two algebraic numbers and add them, it’s not immediately clear that this sum can be expressed as a root of a polynomial with integer coefficients. It turns out to be the case, though, and similarly for subtraction, multiplication and division.

Here’s are some examples. Both \sqrt{2} and \sqrt{3} are algebraic numbers, being the roots of X^2 -2 and X^2 - 3, respectively. Now of course \sqrt{2}\sqrt{3} = \sqrt{6} is a root of the polynomial X^2 - 6. So in this case the product of these two algebraic numbers is again algebraic. The quotient is also algebraic: \sqrt{2}/\sqrt{3} is a root of the polynomial 3x^2 - 2. The most “difficult” part is to show that \sqrt{2} + \sqrt{3} is algebraic. You can check for yourself that it’s a root of X^4 - 10X^2 + 1. I suggest trying to compute a polynomial with integer coefficients which has \sqrt[3]{2} + \sqrt{3} as a root. You can proceed by considering a polynomial of degree 6 of the form X^6 + ax^5 + bX^4 + cX^3 + dX^2 + eX + f, where a, b, c, d, e, f are variables. Then plug in \sqrt[3]{2} + \sqrt{3} and try to solve for the variables.

Ok, so algebraic numbers can be added, subtracted, etc. We also noted last time that every rational numbers is algebraic. Of these, two play a particularly central role. These are the numbers 0 and 1. They are special since they are additive and multiplicative identities, respectively. This just means that 0 added to anything does not change it, and also multiplying by 1 does not change things. These are the arithmetic structures that we’ll want our symmetries to preserve.

Let’s be a bit more formal and give a precise definition. A symmetry of the algebraic numbers is a map f \colon \mathbb{A} \to \mathbb{A} which satisfies the following properties:

  1. It preserves zero: f(0) = 0.
  2. It preserves one: f(1) = 1.
  3. It preserves addition: f(x + y) = f(x) + f(y) for all algebraic numbers x and y.
  4. It preserves multiplication: f(xy) = f(x)f(y) for all algebraic numbers x and y.

It’s a formal consequence of these few properties that a symmetry of \mathbb{A} also preserves subtraction and division. Recall that the symmetries of an object in the plane are those symmetries of the entire plane which preserve the object as well. If \alpha is an algebraic number, then in analogy we define the symmetries of \alpha to be all those symmetries of \mathbb{A} which fix \alpha. That is, a symmetry of \alpha is a map of the ambient space (analogy: map of the plane) f \colon \mathbb{A} \to \mathbb{A} preserving all the arithmetic structures discussed above (analogy: distance preserving), and with the added feature that f(\alpha) = \alpha (analogy: it fixes the object in the plane which is under consideration).

I want to end by showing that if f \colon \mathbb{A} \to \mathbb{A} is a symmetry of the ambient space, then it fixes every rational number. We’ll start by showing that f must fix every positive integer. If \alpha is a positive integer, then we can write \alpha = 1 + 1 + \cdots + 1 for some numbers of ones. Then since f preserves addition and one, we deduce that

f(\alpha) = f(1 + 1 + \cdots + 1) = f(1) + f(1) + \cdots + f(1) = 1 + 1 + \cdots + 1 = \alpha.

So f fixes every positive integer \alpha. To see that f fixes every negative integer as well, we have to use the fact that f respects subtraction. To see that f fixes every rational number, let \alpha be rational and write \alpha = a/b for two integers a and b. Then since f respects division we get

f\left(\alpha\right) = f\left(\frac{a}{b}\right) = \frac{f(a)}{f(b)} = \frac{a}{b} = \alpha.

This shows that f fixes every rational number (although it need not fix every algebraic number). So every symmetry of the ambient space \mathbb{A} is a symmetry of each rational number. You can thus think of them as symmetries of the rational numbers. The collection of all such symmetries is denoted \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) and called the absolute Galois group of the rational numbers. It is one of the most mysterious and beautiful objects in mathematics.

That’s it for now, but expect one more post on this topic. I’ve sort of left you hanging with a definition of arithmetic symmetry, but I haven’t at all given any indication about its importance. In the next post on this subject I’ll try and explain just why mathematicians care about the absolute Galois group of the rationals, and hence why anyone would care that numbers have symmetries. It’ll take me some time to think about the best way to communicate some of the cool facts about \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) in the most elementary way possible, so expect another pause before I write on this topic again.


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