Big Rings Inside Number Fields

The discussion of big rings requires a review of a few more definitions. As above K is a number field.

Any prime ideal p of OK is maximal and the residue classes of OK modulo p form a field. This field is always finite and its size (a power of a rational prime number) is called the norm of p denoted by Np.

• If W is a set of primes of K, its natural density is defined to be the following limit if it exists:

• Let K be a number field and let W be a set of primes of K. Let OKbe the following subring of K.

If W = 0, then OK= OK —the ring of integers of K. If W contains all the primes of K, then OK = K .If W is finite, we call the ring small (or the ring of

W-integers). If W is infinite, we call the ring big. These rings are the counterparts of the “in between” subrings of Q.

• Given a field extension M/K and a prime ideal pK of K, we can talk about factorization of pK in M. In other words when we look at the ideal pK OM of OM, it might not be prime any more but a product of prime ideals of OM. So, in general, in OM we have pK = П?=1 pM i, where pMд,pM,k are distinct prime ideals. We will call ideals of OM occurring in the factorization of a prime ideal of

OK conjugate over K and note the following property of the conjugate ideals. If x e OK c OM and ordPMix < 0 for some i, then ordPMx < 0 for all j.

Before discussing HTP and definitions of Z over big subrings of number fields, we should note that small subrings of number fields are also covered by the results of J. Robinson. To be more precise we have the following proposition (see [40]).

Proposition 3.7 Let K be a number field and let pK be a prime of K. In this case the set {x e KordPKx > 0} has a Diophantine definition over K

Now taking into account the fact that the set of non-zero elements has an existential definition over all small and big rings of any number field, we have the following corollary.

Corollary 3.5 Let K be a number field and let SK be a finite set of primes ofK. In this case, OK has a Diophantine definition over OK,Sk.

Thus in all cases where we know HTP to be undecidable over the ring of integers of a number field, we also have that HTP is undecidable over any small subring of the field.

We now move on to big subrings. The main difference between the big subring situation over Q and over number fields is that we were able to construct a Diophantine definition of Z over some big subrings of non-trivial extensions of Q. We describe these rings below.

Theorem 3.9 Let K be a number field satisfying one of the following conditions:

  • K is a totally real field.
  • K is an extension of degree 2 of a totally real field.
  • There exists an elliptic curve E defined over Q such that [E (K) : E (Q)] < to.

Let e > 0 be given. Then there exists a set S of non-archimedean primes of K such that

  • The natural density of S is greater 1 — ^ 1 q] - e.
  • Z is Diophantine over OK.
  • HTP is unsolvable over OK.
  • (See [52, 53, 55, 58, 59].)

One immediately notices that all the fields to which our theorem applies are the fields where we have definitions of Z over the rings of integers. This is not an accident of course. Given a number field extension M/K and an integrally closed subring R of M, call the problem of defining R П K over R a “vertical” problem and call the problem of defining R over M a “horizontal” problem. (So a vertical problem involves an algebraic extension and a horizontal problem does not involve algebraic extensions, i.e. everything takes place inside the same field.) Using these terms, one could say that we learned how to solve a vertical problem over the fields mentioned in Theorem 3.9 and, using an observation concerning conjugate prime ideals, one can adapt these vertical solutions for horizontal purposes. In other words, let K be a number field and let WK be a collection of prime ideals of K with the following property: all but finitely many ideals in WK have a distinct conjugate over K such that this conjugate is not in WK. In this case OK,Wk П Q = Oq,sk , where SK is either finite or empty and thus either Oq,Sk = Z or Z has a Diophantine definition of Oqsk . So to define Z over OK,Wk with this type of WK it is enough to define OK,Wk П Q, that is to solve a vertical problem.

Relative to solving the corresponding vertical problem over the ring of integers, over OKWk there are some additional difficulties related to bounding of heights, but the overall design of the weak vertical method is unchanged. One should note that by construction the density of the set of the inverted primes can never be one. To get results concerning big rings where the density of inverted primes is one we need Poonen’s method and an elliptic curve of rank one.

There are various generalizations of Poonen’s theorem to number fields. However the situation is more complicated over a number field and instead of constructing a model of Z by “approximation”, what is constructed there is a model of a subset of the rational integers over which one can construct a model of Z. In short, one constructs a “model of a model” (see [36].) There are also analogs of Theorems 3.5 and 3.6 (see [11, 60]). As in the case of the rings of integers, these big ring results extend to all number fields but only conjecturally depending as they are on Shafarevich-Tate conjecture. The situation is different, however, with the “other end of the spectrum” results. As we will see below, they extend seamlessly to all number fields. Before we get to those results, we need to say a few words about presentations of number fields, their primes and their big subrings.

So far most of the results we have discussed above concerning number fields are definitional in nature and do not require a discussion of the presentation of the object involved, just the language in which the definitions are made. The language of course is the language of rings, possibly with finitely many additional constants. However, when we start talking about undecidability we do need to worry about how the objects are presented. Of course number fields and rings of integers have very easy, naturally computable presentations in terms of an integral basis over Q. (If we choose an integral basis, then the ring of integers can be generated as a Z-module from the basis.) The situation becomes more complicated when we discuss big subrings (or even small subrings). Big subrings of Q are computable inside a standard presentation of Q precisely when the set of primes allowed in the denominator is computable. If the set is c.e. but not computable, then as we pointed out above, the ring has a computable presentation, just not as a part of the computable presentation of Q.

The situation in the big subrings of number fields is similar since we have a computable way to describe the primes of a number field. If a number field K is given by the minimal polynomial of its generator (inside a computable presentation of Q, this generator can be given explicitly), and we choose a rational prime p, then within the standard computable presentation of K, using the power basis of the field generator, we can algorithmically determine the number of distinct factors p has in K. Further for each factor we can effectively find an algebraic integer such that this integer has order one at this factor but not at any other factors of p. For a factor p of p let a(p) be this algebraic integer. Now we can represent the prime p by the pair (p, a(p)), where a(p) is given by its coordinates with respect to the fixed basis of K over Q. Further, given an element x of K, we can effectively compute n(x) and d(x) in terms of our presentation of primes and assuming that WK is a computable set of primes, we can determine whether x e OKwk .

We can re-use the identification of HTP of a particular ring with a subset of positive integers containing the indices of all polynomials with coefficients in the ring having a root in the ring. Our next observation is that HTP(K) <T HTP(Q) since we can rewrite any polynomial equation over K with variables ranging over K as a system of polynomial equations over Q with variables ranging over Q. Further, as over Q, we also have for any WK that HTP(OKHTP(K). Finally, we also have the following results from [12].

Theorem 3.10 For any number field K there exist a c.e. set Wk of primes of K of lower density equal to zero such that HTP(Ok,Wk ) =T HTP(K) <T HTP(Q) and WK <THTP(K).

Theorem 3.11 For any number field K and any positive integer m there exist sets W,...,'Wm of primes of K such that W <THTP( K), each Щ has a lower density zero and W U---U Wm is a partition of all primes of K.

As over Q we can also also ask what do we know about definability and decidability using the full first-order theory.

 
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