The Galois Correspondence

30 November 2009

Lately we’ve been talking about fields, which are structures made of numbers, and groups, which are structures made of symmetries; we recently established a connection between the two in the form of Galois groups.  To see the full beauty of the theory, we need to add one last layer of complexity.  Today we will look at so-called intermediate fields; we will see there is a close relationship between the arrangement of intermediate fields and the structure of the Galois group.

An Example

Consider the field \mathbb{Q}(\sqrt{2},\sqrt{3}), which is obtained by starting from the rational numbers and attaching the square roots of 2 and 3.  It turns out this is just the set of  numbers of the shape a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}, where a,b,c,d are fractions.  (It might not be totally obvious that this thing is a field, but it is.)  We’ll be thinking of this as an extension of the smaller (and more familiar) field of fractions \mathbb{Q}.

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Symmetries of Number Systems

19 November 2009

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.


Number Systems

5 November 2009

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 (a+b=b+a)
  • addition is associative (a+(b+c)=(a+b)+c)
  • 0 is the additive identity (a=0+a)
  • subtraction is the “opposite of addition” in the sense that a + (b-a) = b
  • multiplication is commutative (ab=ba)
  • multiplication is associative (a(bc)=(ab)c)
  • 1 is the multiplicative identity (a=1a)
  • division is the “opposite of multiplication” in the sense that a (b/a) = b, assuming a\neq 0.
  • multiplication distributes over addition (a(b+c)=ab+ac)

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 1/0.  Why not?  Why was division by zero so much harder than square roots of negative numbers.  Notice by the way that, if any a is any number whatsoever in any conceivable field, then the distributive law a\cdot 0= a(1-1)=a\cdot 1 - a\cdot 1 = 0.  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

  1. \mathbb{Q}, the field of all rational numbers
  2. \mathbb{R}, the field of all rational numbers
  3. \mathbb{C}, the field of all complex numbers
  4. \mathbb{Z}_{11}, the field of integers modulo 11 (convince yourself that division really works here!)
  5. \mathbb{Q}(i), the field of all complex numbers \frac{l+mi}{n} (l,m,n 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.

  1. \mathbb{Z}, the integers
  2. \mathbb{Z}_{12}, 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 a+ib, where a,b are real.  Whenever there is a finite list of elements \{\alpha_1,\alpha_2,\ldots,\alpha_k\} and every element in the larger field has a unique expression in the form a_1\alpha_1+\cdots+a_k\alpha_k, we say that the larger field is a finite extension, and k is the degree of the extension.  So the complex numbers are a degree-2 extension of the real numbers (and the natural basis is \{1,i\}).

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 \mathbb{C} arises, by beginning with \mathbb{R} and adding solutions to x^2+1=0.  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 E is a finite extension of F, then there is some polynomial p(x)=0 over F such that E is obtained by adding a root of p(x)=0 to F.

Finite extensions of \mathbb{Q} 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 F is a field and E is some field containing F, then we will want to study the symmetries of E over F?

So what, exactly, is the right thing to mean by a symmetry of a number system?  In the case of \mathbb{C} over \mathbb{R}, 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.


Thought Experiment: Talking to the Other Aliens

22 October 2009

This is a direct continuation of the previous post, so read that one first if you haven’t yet.  In some sense this post is simpler than the previous, in that it uses simpler concepts and doesn’t involve understanding of the real number system.  But it may be harder for many readers, because I’m asking you to imagine an alien race which does not understand certain things that you probably can’t remember a time when you didn’t understand.  And it’s hard to imagine what it would be like not to know what we know.

It’s interesting, isn’t it, how people are much better at temporarily adding an unfamiliar concept to their working context than they are at temporarily subtracting a familiar one?

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Thought Experiment: Talking Math with the Aliens

20 October 2009

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 i?  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 i?  After all, (-i)^2=-1 too.  Maybe blarg is negative i?  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 i or -i.

(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 i 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 (3+2i)+(4-i)=6+i and we both think (3+2i)(4-i)= 14+5i.  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 a+bi is a root, then so is a-bi.  Now we see the reason that underlies this truth: no algebraic statement in terms of real numbers can distinguish a\pm bi from one another.  The point in your mind I call a+bi, I call a-bi, 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 a+bi and a-bi 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.