Symmetries of Number Systems

I know it’s been a long time since my last Galois Theory post; my apologies.

When last we met, we had introduced two important algebraic structures: the group and the field. At least superficially these are similar mathematical concepts, in that both are sets of things with operations that satsify certain rules, but we aren’t thinking of them the same way. We think of the objects in a group as symmetries (and the operation is function composition); we think of the objects in a field as numbers (and the operations as addition and multiplication). (See the last two posts for a refresher, or else see the corresponding wikipedia articles for an overflow of information.)

What’s missing in this story is the interaction between these structures. That is, what are symmetries of a number system? The answer is easy to state: a symmetry is any permutation of the numbers in the field which preserves addition and multiplication. That is, it is a function \phi from F to F such that \phi(x+y)=\phi(x)+\phi(y) and \phi(xy)=\phi(x)\phi(y). Saying that the symmetry preserves the basic operations is the same as saying that the symmetry preserves any possible statement in the language of basic algebra. These symmetries have a fancy name: field automorphisms.

Whenever you have a symmetry, it’s interesting to think about about which objects correspond to themselves under the symmetry. In geometric contexts, you already know this: when you think of a reflection, you think first of the axis of reflection, and when you think of a rotation, you think first of the point about which you are rotating. And any symmetry of a field has to leave certain numbers alone. If \phi is an automorphism of any field, we must have \phi(0)=0, because 0+0=0, so \phi(0)+\phi(0)=\phi(0) and no other number is like that. Likewise, \phi(1)=1, and then we have \phi(p/q)=p/q for any rational number p/q. That is, every symmetry of every field has to fix the rational numbers.

Remarks on the Definition

  1. Am automorphism of any algebraic structure—a group, a field, a ring, a module, etc.—is just a permutation of its elements which preserves the structure. So a group automorphism is a permutation of the group elements which preserves group composition. (So symmetry groups have their own symmetry groups…wild.)
  2. Actually, automorphism groups are themselves a special case of a more general template for constructing interesting groups. If you have a configuration of objects, the collection of all the permutations of the objects which preserve some interesting properties is automatically a group in a natural way.
  3. Every symmetry must fix the rational numbers, at least; that is, a symmetry might happen to leave even more numbers alone. This, it will turn out, is where things really get interesting.
  4. I’m implicitly assuming that the field we’re talking about has a copy of \mathbb{Q} inside it. This is true about most fields that you probably care about; more precisely it is true for any field where repeatedly adding up 1’s does not ever give 0. In more exotic cases, so-called fields of positive characteristic, the role of \mathbb{Q} is played by some other simple field. For ease of language, I’ll gloss over this point.
  5. Symmetries preserve rational numbers because they can be expressed in terms of the basic operations and 0 and 1, and 0 and 1 can themselves be completely described in terms of the operations. This should remind you of the alien posts; essentially this is the reason why it’s always straightforward to establish shared language for the rational numbers.

Now, it might happen that we want to insist that an automorphism of a field E fix some smaller field F. In the first alien post, we were able to establish the real number system by bringing in some other concepts, so the only symmetries of the complex numbers that made sense in context were the ones that preserve the real numbers. We call this the symmetry group of E relative to F.

The group of symmetries of a field E relative to a subfield F is called the Galois group G(E/F). (The slash is read as “over”, not “divided by”.) If you don’t operate relative to some subfield (which really just means you are operating over the prime field), that’s called the absolute Galois group G(E) or G(E/\mathbb{Q}).

There is a direct connection between the Galois group and the alien thought experiment. If two parties have possibly different languages for talking about the field E, but they know they agree on the smaller field F, then each element of the Galois group G(E/F) corresponds to a way of matching up their vocabularies for E compatibly with F.

In some sense, then, the larger the Galois group is, the more complicated the extension is. But we can be a great deal more explicit than this. In the next installment we’ll see that, under the right circumstances, there is a very precise correspondence (the so-called Fundamental Theorem of Galois Theory) between the structure of the Galois group and the structure of a field extension.

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