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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.
I have been reading Joseph Silverman’s new book on arithmetic dynamics lately. There’s a lot of really fascinating stuff in there, including a large number of potential research problems that are currently way beyond me, but I’ll continue thinking about them! Most interesting so far is the Uniform Boundedness Conjecture:
Let , , and be integers. Then there exists a constant such that for any number field with and any morphism of degree , the number of preperiodic points of in is at most .
Not much is known about this conjectures; even the case , , and is open. It’s even open if we restrict to morphisms of the form . Bjorn Poonen has shown, however, that these maps have no rational periodic points of exact period 4 or 5; it is conjectured that they have no rational periodic points of exact period greater than 3.
However, there is a positive result of the above type that doesn’t depend that much on some of the above quantities:
Let be a number field and be a rational map over . Let and be prime ideals of so that has good reduction at and (meaning that when we reduce modulo and , we end up with a map of the same degree as ) and whose residue characteristics are distinct. Then the period of any periodic point of in satisfies , where denotes the (absolute) norm.
(See, for instance, my algebraic number theory notes for definitions of some of these terms.)
Anyway, that wasn’t really the point of this post, as you may have guessed from the title. I meant to talk about theorems that pretend not to be related to dynamical systems but actually are. First we need to discuss height functions a bit; there’s a lot more about them in Silverman’s book and in my elliptic curve notes.
We let be a number field and the set of standard absolute values on (These are the absolute values on whose restriction to is either the standard absolute value or one of the -adic absolute values.) We write (where denotes the completion of with respect to the absolute value ). Suppose ; we can then write for some . We then define the height of with respect to to be . One can check that this is well-defined, and that if is a finite extension of number fields and , then . Hence it is possible to define the absolute height of by .
One of the key facts about heights is the following: If and are constants, then is finite. A corollary is the following well-known result of Kronecker:
Let be nonzero. Then if and only if is a root of unity.
Proof: If is a root of unity, then is clear. Now suppose that . For any and , we have , so , so is a set of bounded height and is therefore finite. Therefore there are integers such that , so is a root of unity.
And now for linear algebra. Sheldon Axler has a well-known book on linear algebra without determinants. He therefore uses dynamical systems to show the following familiar result:
Theorem: Every operator on a finite-dimensional nonzero complex vector space has an eigenvalue.
Proof: Let be such a vector space of dimension . Let be an operator on , and let be nonzero. Then cannot be a linearly independent set. Hence there exist , not all zero, so that . Suppose is maximal with respect to . Then . Since we’re working over , the polynomial factors as . We then have . Hence some is not injective. This is an eigenvalue for .