We begin with a few definitions.
Definition 1: A integral domain is called a Dedekind domain if it is noetherian, every nonzero prime ideal is maximal, and it is integrally closed in its field of fractions.
Definition 2: A ring is called a discrete valuation ring (DVR) if it is a principal ideal domain (PID) with exactly one nonzero prime ideal. (In other language, a DVR is a local PID of Krull dimension 0 or 1.)
One very important property of Dedekind domains is that ideals have unique factorizations as products of prime ideals. I used this property in the case of rings of integers in my last post to say that if is an extension of number fields with rings of integers , so if is a prime ideal, then we can write . But this result holds in more generality, for any Dedekind domain.
Also, it is very easy to check that a DVR is a Dedekind domain. But one very common occurrence of DVRs is as localizations of rings of integers. In particular, if is a Dedekind domain and is a prime ideal of , then is a DVR.
One way to interpret a DVR is through the following filtration of ideals. Let be a DVR, and let be the unique nonzero prime ideal of . Then every nonzero ideal of is of the form for some (where by I mean ). Now, for any , there is an integer so that . We can now define a function (where includes 0 in this case) by as above. We can extend our definition of to all of by setting .
It is also possible to extend to the quotient field of by setting ; it is easy to check that this is well-defined. Now, satisfies the following properties:
1) is a surjective homomorphism.
We call such a function a valuation of the field .
Knowing is enough to reconstruct , since . Furthermore, . We call the valuation ring of .
Let’s look at a few places where DVRs arise naturally.
1) As we mentioned earlier, the localization of a Dedekind domain at a prime ideal is a DVR. So, for example, is a DVR if is a prime. The unique prime ideal is .
2) The ring of -adic integers is a DVR with unique prime ideal . Also, finite extensions of the field of -adic numbers inherit valuations from , and so they contain DVRs as described above. In particular, if is a finite field extension, then the integral closure of in is a DVR.
Now, if is a DVR and is its prime ideal, then is a field. In the cases described above, this will always be a finite field; in what follows, we always assume that this field is finite. We call the residue field.
We can also put a topology on a valued field by letting the following sets be a basis for the topology: if and is an integer, then is an open set. These sets generate the topology. In what follows, we will assume that is complete as a topological space with this topology. Finite extensions of are complete with respect to this topology, so this will be our motivating example. The residue fields will also be finite.
Last post, I pointed out that if is a Galois extension of number fields, then . This holds more generally, however. If is a finite Galois field extension, and is a Dedekind domain so that is the quotient field of , and is the integral closure of in , then we still have .
We now interpret this in the case of a field complete with respect to a discrete valuation , and the valuation ring of . Let be a finite Galois extension, and let be the integral closure of in , or, equivalently, the valuation ring of . Then is also complete with respect to a discrete valuation that is very closely related to , as we will see soon.
Let be the prime of , and let be the prime of $B$. Since there is only one prime, . Hence . Now, if , then , and if , then . (But we won’t need these results in what follows, at least today.) The implication is the decomposition group of the extension is the entire Galois group.
We can put a filtration on the Galois group as follows: For , let . We call the ramification group of . is the entire Galois group (or the decomposition group; is the inertia group. Also, each is normal in .
Now, I won’t prove it here, but it can be shown that if the residue field is finite of characteristic and is complete, then for each , is a direct product of copies of , and is a subgroup of the roots of unity of (and hence finite and cyclic of order prime to ). Hence, by basic group theory or otherwise, is a semidirect product of a normal Sylow -subgroup and a cyclic group of order prime to . In particular, is solvable. However, as shown in the last post, is cyclic since it is the Galois group of an extension of finite fields. Hence:
Theorem: is solvable.