First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers

Abstract

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold. (1) For any rational prime q and any positive rational integer m, algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by q m . (2) Given a prime q, and an integer m > 0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set {ξ p ℓ |ℓ ∈ Z >0 , p = q is any prime such that q m+1 |(p − 1)}. (3) The first-order theory of any abelian extension of Q with finitely many ramified rational primes is undecidable. We also show that under a condition on the splitting of one rational prime in an infinite algebraic extension of Q, the existence of a finitely generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is indecidable. produced a uniform definition of Z over rings of integers of number fields, R. Rumely in [30] improved J. Robinson's result making the definition of the ring of integers over number fields uniform across fields. More recently, B. Poonen in [22] and J. Koenigsmann in [12] updated J. Robinson's definition of integers by reducing the number of universal quantifiers used in these definitions, B. Poonen to two and J. Koenigsmann to one.The desire to reduce the number of universal quantifiers is motivated to large extent by the interest in extending Hilbert's Tenth Problem to Q. This would be accomplished by a purely existential definition of Z over Q. Unfortunately there are serious doubts as to whether such a definition exists. See [6], [21] and [33] for surveys on Hilbert's Tenth Problem and related questions of definability.A lot of work aiming to prove the decidability of the first-order theory has centered around various infinite extensions of Q. (See [6] for a survey of these results.) One of the more influential results was arguably due to R. Rumely in [31], 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 [40] that the first-order theory of this ring was decidable. Another remarkable result is due to M. Fried, D. Haran and H. Völklein in [10], where it is shown that the first-order theory of the field of all totally real algebraic numbers is decidable. This field constitutes a boundary of sorts between the "decidable" and "undecidable", since J. Robinson showed in [27] that the first-order theory of the ring of all totally real integers is undecidable. In the same paper, she also proved that the first-order theory of a family of totally real rings of integers is undecidable and produced a "blueprint" for such proofs over ring...

Undecidability, unit groups, and some totally imaginary infinite extensions of $\mathbb{Q}$

“…Clearly, if O L is definable in L and the first-order theory of O L is undecidable, then the first-order theory of L is also undecidable. There are many results on the definability of rings of integers in infinite algebraic extensions of Q; see, for example, the work of Fukuzaki [7], Shlapentokh [22], and Videla [27]. The following result of Shlapentokh, presented in [22,Example 4.3], will suffice for our purposes in this paper.…”

“…There are many results on the definability of rings of integers in infinite algebraic extensions of Q; see, for example, the work of Fukuzaki [7], Shlapentokh [22], and Videla [27]. The following result of Shlapentokh, presented in [22,Example 4.3], will suffice for our purposes in this paper. §3.3], gives a sufficient condition for when a ring of integers has undecidable first-order theory.…”

“…In this theorem, the ring of integers O K is undecidable by a result of Vidaux and Videla [26]. This fact is used to deduce the undecidability of O L , and therefore the undecidability of L because O L is definable in L by a result of Shlapentokh [22]. The strategy, as in the case of Q (2) , is to exploit the unit group O × L to define a sufficiently large subset W of O L which contains only totally real elements.…”

“…Vidaux and Videla expanded on her ideas to prove the undecidability of a large class of rings of integers O K in totally real fields K [25,26]. When combined with definability results, such as those of Fukuzaki [7], Shlapentokh [22], and Videla [27], this proves the undecidability of many totally real fields.…”

“…There are many results which partially answer this question and demonstrate that some infinite algebraic extensions of Q have undecidable first-order theory, while others do not. For example, Shlapentokh proved that every abelian extension of Q which is ramified at only finitely many primes is undecidable [22,Theorem 5.5], generalizing a result of Videla [28]. On the other hand, recall that an algebraic number is said to be totally real if its minimal polynomial has only real roots.…”