Infinite Extensions

In this section we want to discuss some of the things we know about infinite algebraic extensions of Q and point out the differences and similarities with the finite extension case. We start with a description of the global picture to the best of our understanding.

Let Q be a fixed algebraic closure of Q and consider a journey from Q to its algebraic closure, passing through the finite extensions of Q first, then through its infinite extensions fairly “far” from the algebraic closure, and finally through the infinite extensions of Q fairly “close” to Q.

As we get closer to Q, the language of rings looses more and more of its expressive power, i.e. sets which were definable before (in either full first-order language or existentially) would become undefinable and simultaneously some problems which were undecidable before would become decidable. For the author the ultimate goal of the infinite extension investigation in this setting is to describe this transition. Of course, the completion of this project is probably far away. The boundary (in terms of extensions of Q) where previously undecidable things become decidable (e.g. the first-order theory of fields) and previously definable things become undefinable (e.g. rings of integers over their fields of fractions using the full first-order language) is likely to have a very complex description.

Further, the decidability issue is muddled by the following aspect of the problem which does not manifest itself over finite extensions. It can be shown that an algebraic extension of Q with a decidable existential theory (a fortiori a decidable first-order theory) must have an isomorphic computable copy inside a given algebraic closure of Q. (See [18].) Thus, a field can have an undecidable theory (existential or elementary) simply because it has no decidable conjugate (under the action of the absolute Galois group) and not because of, should we say, “arithmetic” reasons. We are tempted to call such fields as having a “trivially” undecidable theory.

A simple example of a field with a trivially undecidable theory can be described as follows. Consider a computable sequence of prime numbers {pi} and choose an undecidable subset A of Z>0. Now let K be an algebraic extension of Q formed by adding a square root of every pi such that i e A. It is not hard to see that this field is not computable as a subfield of Q, but if A is c.e. it is computably presentable. It is Galois and has no other conjugates besides itself. So by the argument above the existential theory of this field (in the language of rings) is undecidable, but surely this is not a very interesting case. A related point which should be made here is that if we consider uncountably many isomorphism classes of fields, then “most” of them will have undecidable theories simply because we have only countably many decidable theories in the language of rings.

Having moved the trivial considerations aside and concentrating on computable fields, we discover a relatively patchy state of knowledge concerning definability and decidability. If we look at the fields “close” to the algebraic closure, we see a number of decidability results. Here, of course, the problem concerning the full first- order theory is the more difficult one as compared to the problem of the existential theory. One of the more influential decidability results is due to Rumely [46], where he showed that Hilbert’s Tenth Problem is decidable over the ring of all algebraic integers. This result was strengthened by L. van den Dries proving in [63] that the first-order theory of this ring was decidable. Another remarkable result is due to Fried et al. [15], where it is shown that the first-order theory of the field of all totally real algebraic numbers is decidable. This field quite possibly is a part of the decidability/undecidabilty boundary we talked about above, since J. Robinson showed in [41] that the first-order theory of the ring of all totally real integers is undecidable. Together these two results imply that the ring of integers of this field is not first-order definable (in any way) over the field.

Among other famous decidability results is the result due to A. Prestel who building on a result of A. Tarski showed that the elementary theory of the field of all real algebraic numbers is decidable (see [38, 62]). Further, due to Yu. Ershov, we know that the field of all S-adic algebraic numbers is decidable provided S is a finite set of rational primes. (The field of all S-adic algebraic numbers is the intersection of all Qn Q p, p e S with Q being some fixed algebraic closure of Q. See [13].) The rings of integers of the fields of real and p-adic algebraic numbers are decidable too. (See [37].)

We now turn our attention to definability and undecidability results. We have already mentioned a well-known result of J. Robinson proving that the ring of integers of the field of all totally real integers is undecidable. In the same paper, J. Robinson also outlined a plan for showing undecidability of families of rings of integers. Using some of these ideas, their further elaboration by C.W. Henson (see [63, p. 199]), and R. Rumely’s method for defining integrality at a prime, C. Videla produced the first-order undecidability results for a family of infinite algebraic extensions of Q in [64-66]. More specifically, C. Videla showed that the first-order theory of some totally real infinite quadratic extensions, any infinite cyclotomic extension with a single ramified prime, and some infinite cyclotomic extensions with finitely many ramified primes is undecidable. C. Videla also produced the first result concerning definability of the ring of integers over an infinite algebraic extension of Q:he showed that if all finite subextensions are of degree equal to a product of powers of a fixed (for the field) finite set of primes, then the ring of integers is first-order definable over the field.

In a recent paper [17], K. Fukuzaki, also using R. Rumely’s method, proved that a ring of integers is definable over an infinite Galois extension of the rationals such that every finite subextension has odd degree over the rationals and its prime ideals dividing 2 are unramified. He then used one of the results of J. Robinson to show that a large family of totally real fields contained in cyclotomics (with infinitely many ramified primes) has an undecidable first-order theory.

In another recent paper (see [47]), the author attempted to determine some general structural conditions allowing for a first-order definition of the ring of integers over its fraction field over infinite algebraic extensions of Q. As we speculated above, a definitive description of such conditions is probably far away, but one candidate is the presence or the absence of what we called “q-boundedness” for all rational primes q. We offer an informal description of this condition below.

Given an infinite algebraic extension Kinf of Q we consider what happens to the local degrees of primes over Q as we move through the factor tree within Kinf. A rational prime p is called q-bounded if it lies on a path through the factor tree in Kinf where the local degrees of its factors over Q are not divisible by arbitrarily high powers of q .If every descendant of p in every number field contained in Kinf has the same property, then we say that p is hereditarily q-bounded.

For q itself we require a stronger condition: the local degrees along all the paths of the factor tree should have uniformly bounded order at q .If this condition is satisfied, we say that q (or some other prime in question) is completely q-bounded. If all the primes p = q are hereditarily q-bounded and q is completely q-bounded, we say that the field Kinf itself is q-bounded, and we show that the ring of integers is definable in such a field. Rings of integers are also definable under some modifications of the q-boundedness assumptions, such as an assumption that all primes p = q are hereditarily q-bounded and q is completely t-bounded for some prime t = q, etc.

It is not hard to see that the fields considered by C. Videla and K. Fukuzaki are in fact q-bounded. As mentioned above, C. Videla’s results concerned infinite Galois extensions of number fields, where all the finite subextensions are of degree divisible only by primes belonging to a fixed finite set of primes A. Consequently, in the fields considered by C. Videla all the primes are completely q-bounded for any q e A, and thus all these fields are certainly q-bounded. K. Fukuzaki’s fields are 2-bounded. However, the examples constructed by C. Videla and K. Fukuzaki do not exhaust all the q-bounded fields. One example not covered by these authors is any field where for some fixed rational prime q and some fixed m e Z>0 we can adjoin to Q all tn-th roots of unity for any positive integer n and for any rational prime t such that qm does not divide t — 1.

We suspect that q-boundedness or a similar condition is necessary for definability of the ring of integers. While non-definability examples are scarce over infinite extensions, we offer the following ones: the field of all totally real numbers is not q-bounded and as we mentioned above has the ring of integers not definable over the field. Further, the field of real algebraic numbers is also not q-bounded and its ring of integers is not definable over the field by a result of Tarski [62].

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