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 , 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 , where 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 .

So what are the intermediate fields? That is, what fields containing are contained inside our big field ? We are looking for collections of numbers in which contain all of and which support addition, subtraction, multiplication, and division except by zero. It might seem like there would be a *lot* of such objects, but it turns out there are five in all.

**Intermediate Fields**

- the big field
- , which is just the numbers of the form , where $a,b$ are fractions.
- , which is just the numbers of the form , where $a,c$ are fractions.
- , which is just the numbers of the form , where $a,d$ are fractions.
- the base field

At first you might expect there to be a lot more. For example, you might try adjoining all kinds of strangely chosen numbers. Fields like ,, or would be good candidates, but it turns out each of these is just another way of describing a field already on this list.

What about the Galois group? It turns out to have an easy description. There are only four symmetries to consider.

- , the trivial symmetry which maps any element to itself,
- , the symmetry which maps any element to
- , the symmetry which maps any element to
- , the symmetry which maps any element to

(If you’re feeling a little ambitious, check that these really are symmetries; if you’re feeling more than a little ambitious, check that this is a complete list!)

What are the subgroups of this group? That is, what combinations of these symmetries form a group in their own right? With only four elements to work with, it’s no surprise that there aren’t that many combinations that work. Here’s a complete list (you should be able to work this out for yourself without much trouble).

**Symmetry Subgroups**

- the trivial subgroup, just containing
- the two-element subgroup
- the two-element subgroup
- the two-element subgroup
- the full Galois group .

Right away you might guess that something is up, because this list is the same length as the list of intermediate fields. Of course, I probably wouldn’t be much of a mathematician if I got excited every time I saw a list of five things. The connection is mucb deeper than that. Each intermediate field consists of exactly those numbers which are left unchanged by the symmetries in the corresponding subgroup. (You may recall that in the last Galois theory post, I pointed that the single most important feature of a symmetry is usually which things it leaves unchanged.)

For example, look at the objects in position #4. Of all the numbers of the form , the ones which will not be changed by or are precisely those for which . In other words, this subgroup fixes precisely the subfield , which is just the numbers of the form , where $a,d$ are fractions.

So it appears that, by relating each symmetry subgroup to the collection of numbers it doesn’t change, we establish a structural parallel between the intermediate fields and the subgroups of the Galois group.

### The Bottom Line

Superficially there is nothing all that special about this setup. Any time you have a mathematical structure with a natural notion of symmetry and a natural notion of subobject, we can do this. For every subobject, we can ask what subgroup of the symmetries preserve the subobject, and for every symmetry subgroup, we can ask what is the largest subobject fixed by those symmetries. (Loosely speaking, the smaller the subobject, the more symmetries will preserve it, and vice versa.)

This is fine as far as it goes, but there is the potential for things to go awkward. It could happen that all the symmetries fixing a subobject actually turn out to fix a larger subobject. In other words, two subobjects might be fixed by precisely the same symmetries. In such a case, while there is still a sort of connection between the structure of the symmetry group and the deeper structure of the object, it is somehow a distorted representation. Understanding the symmetry group gives you only a blurry notion of the substructure.

Now in our example this “blurriness” did not manifest. The field extension had three intermediate fields (not counting the full field or the base field), and the Galois group had three subgroups (not counting the trivial group or the full group). The correspondence was exact. Did I just pick a lucky example, or there a more important pattern in play here? Amazingly, it turns out that, under the right conditions, the symmetries of fields are magical things. That is, every subfield corresponds to a different group of symmetries. For the magic to work, we need two technical assumptions.

*The extension must be*Our standard way of building larger fields from smaller fields is by adjoining solutions to one or more polynomials that weren’t previously solvable. The extension is normal if we always adjoin**normal**.*all*the solutions of a polynomial whenever we adjoin one.*The extension must be*I don’t really have anything insightful to say about what this means. I said a while back that we could assume that all our fields contain a copy of the rational numbers, and if we only work with that kind of fields, all extensions are automatically separable, so we needn’t worry. For the purposes of this blog, let’s leave it at that.**separable**.

This is the fundamental theorem of Galois theory: for a normal, separable extension , there is a one-to-one correspondence between fields which contain and are contained in and subgroups of . Every symmetry subgroup stabilizes a different field, every field is stabilized by a different symmetry subgroup. Furthermore there is a connection (which can be made precise) between the number of elements in a symmetry subgroup and the complexity of the corresponding subfield. All the important information about the structure of the number system is encoded somehow in the structure of the symmetries, and vice versa.

To repeat: under those two technical assumptions on as an extension of , there is a direct correspondence between the fields in between and and the subgroups of the Galois group . So is this of any use? It’s beautiful, which would be enough for *this* number theorist, but the surprising truth is Galois theory has lots of applications. Lots of questions about fields can be translated . One of the most amazing applications is also one of the first historically: the Abel-Ruffini theorem, which says that there is nothing analogous to the quadratic formula for polynomials of degree higher than 4.

Finding solutions to polynomial equations is one of the fundamental problems of classical mathematics, and it’s not at all obvious why there would be any connection between this problem and the ideas discussed above. In fact, this problem was a primary motivator for the development of group theory in general and Galois theory in particular.

Clearly there is more to this story; stay tuned.