Definability and decidability for rings of integers in totally imaginary fields

We show that the ring of integers of $\mathbb{Q}^{\text{tr}}$ is existentially definable in the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$, where $\mathbb{Q}^{\text{tr}}$ denotes the field of all totally real numbers. This implies that the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$ is undecidable and first-order non-definable in $\mathbb{Q}^{\text{tr}}(i)$. More generally, when $L$ is a totally imaginary quadratic extension of a totally real field $K$, we use the unit groups $R^\times$ of orders $R\subseteq \mathcal{O}_L$ to produce existentially definable totally real subsets $X\subseteq \mathcal{O}_L$. Under certain conditions on $K$, including the so-called JR-number of $\mathcal{O}_K$ being the minimal value $\text{JR}(\mathcal{O}_K) = 4$, we deduce the undecidability of $\mathcal{O}_L$. This extends previous work which proved an analogous result in the opposite case $\text{JR}(\mathcal{O}_K) = \infty$. In particular, unlike prior work, we do not require that $L$ contains only finitely many roots of unity.

1. Introduction 1.1.A motivating example.This paper is motivated by the desire to prove the following theorem concerning definability and decidability.For background and an overview of decidability and definability for infinite algebraic extensions of Q, we refer readers to the introduction of [Shl18]; see also the prequel to our work here [Spr20].Recall that α ∈ Q is totally real if the roots of its minimal polynomial are all real numbers.We write Q tr for the field of all totally real algebraic numbers and Z tr for its ring of integers, and we set i = √ −1.
Theorem 1.1.The ring of integers Z tr of Q tr is existentially definable in the ring of integers of Q tr (i).In particular, the first-order theory of the ring of integers of Q tr (i) is undecidable.
Remark 1.2.The paragraph below Corollary 3.19 in [Koe14] mistakenly declared the ring of integers of Q tr (i) to be decidable. 1The reference given for this claim is [Dar00, Theorem 10.7], which actually proves the decidability of the overring O Q tr (i) [ 1 p ] where p is any prime number.As seen above, the ring of integers O Q tr (i) itself is undecidable.
Theorem 1.1 is proven by extending and generalizing the methods which were first used to prove the first-order undecidability of rings of integers in fields such as Q (2) [MRUV20] and ab for d ≥ 2 [Spr20].Here, F (d) denotes the compositum of all extensions of degree at most d of a field F , and F (d) ab ⊆ F (d) is the maximal abelian subfield.However, notice that the papers [MRUV20,Spr20] additionally proved that the fields Q (d) ab themselves are undecidable by appealing to results of Videla [Vid00a] and Shlapentokh [Shl18] which imply that each Q (d) ab has a first-order definable ring of integers.The situation is entirely the opposite in the setting of Theorem 1.1.Indeed, Fried, Haran, and Völklein proved that Q tr is decidable [FHV94], which implies that Q tr (i) is also decidable.It has been previously observed that, because J. Robinson proved that Z tr is undecidable Date: July 31, 2023. 1 We thank Aharon Razon for pointing this out after reading an earlier draft of this paper.
[Rob62], the field Q tr is thus an example of a subfield of Q whose ring of integers Z tr does not admit any first-order definition in Q tr .In the framework of Shlapentokh [Shl18, §2.1], this is intuitively understood as a result of the fact that the field Q tr is too "close" to Q, so it does not have enough "expressive power" to have a definable ring of integers.By Theorem 1.1, we immediately see that Q tr (i) also has a first-order non-definable ring of integers, as expected.
We pause to note that Q tr (i) is known to be an ω-free PAC field; see [Jar11, Example 5.10.7].Therefore, the non-definability of the ring of integers of Q tr (i) can also be deduced from the following result of Dittman and Fehm: If R is a first-order definable subring of an ω-free PAC field L, then R is a field [DF21, Proposition 3].Their method, which builds upon work of Chatzidakis [Cha19], is entirely different from the techniques that we use below.
1.2.Main results.In general, this paper considers totally imaginary quadratic extensions of totally real fields.We begin with a definability result.Say that a totally real field K ⊆ Q is closed under square roots if α ∈ K whenever α is a totally real number such that α 2 ∈ K.
Theorem 1.3 (Theorem 3.5).If K is a totally real field which is closed under square roots and L is any quadratic totally imaginary extension of K, then This theorem implies Theorem 1.1 immediately, given J. Robinson's proof that Z tr is undecidable [Rob62].However, we can actually prove first-order undecidability for rings of integers O L in a more general context where O K is not known to be existentially definable in O L .For any n ≥ 1, let ζ n denote a primitive n-th root of unity.
Theorem 1.4 (Theorem 4.5).Let S ⊆ N be infinite.Define K 0 = Q({ζ n + ζ n : n ∈ S}) and let K 1 be the maximal totally real subfield of K (2) 0 .If K ⊇ K 1 is a totally real field and L is any totally imaginary quadratic extension of K, then O L is undecidable.
In this theorem, the so-called JR-number is JR(O K ) = 4, i.e., the minimal possible value, while the totally real fields K considered in [MRUV20,Spr20] have the maximal possible value JR(O K ) = ∞; see Section 4.1 for definitions and notation.We also provide a version of our undecidability result, namely Theorem 4.4, which does not place a restriction on the value of JR(O K ).1.3.Proof method: Leveraging unit groups.Within the body of literature concerning undecidability results for rings of integers of algebraic extensions of Q, the totally real subfields of Q and their totally imaginary quadratic extensions have received special attention.There are multiple methods which can be used in this context, but we focus on the work stemming from J. Robinson for now, and delay the discussion of elliptic curves until the following section.
J. Robinson [Rob62] provided a general sufficient condition for proving the undecidability of a ring of algebraic integers O, and we follow Videla [Vid00b] by using an improved version credited to Henson [VDD88]: If there is a parametrized family F of definable subsets of O which contains finite sets of arbitrarily large cardinality, then the first-order theory of O is undecidable.This sufficient condition, along with a theorem of Siegel [Sie21], leads to a strategy that can be applied to O K when K is a totally real field.Indeed, we use the parametrized family of subsets {X t } t∈Q , where X t contains the elements α ∈ O K whose conjugates all lie in the real interval (0, t).Determining whether {X t } t∈Q contains finite sets of arbitrarily large size is related to the JR-number of O K ; see Section 4.1 for more details.This strategy enabled J. Robinson to prove the undecidability of the rings of integers of both Q tr and Q After using this method to prove O K is undecidable for a totally real field K, we wish to do the same for the rings of integers O L for all totally imaginary quadratic extensions L of K. To extend the JR-number argument to such a field, a key ingredient is the fact that ⊆ K is an existentially definable totally real subset of O L .By using sums and difference of powers of units, we can produce a useful parametrized family of subsets of O L analogous to the sets X t defined above.Undecidability is thereby proved for O L in many cases, including ab for d ≥ 2; see [MRUV20,Spr20].However, the assumption #µ(L) < ∞ above implies that this method cannot yet apply when L = Q tr (i).To remedy this, we choose to instead work with the unit group R × of an existentially definable non-maximal order R ⊆ O L with µ(R) = {±1} trivial.In this case, we can show that (R × ) 2 is totally real set.By choosing a suitable ring R ⊆ O L , we obtain the foundation for developing a unit group-based argument similar to the ones appearing in [MRUV20,Spr20] which allows L to contain infinitely many roots of unity.
1.4.A comparison with abelian varieties.We conclude this section by comparing our main theorems to similar results which leverage elliptic curves instead of unit groups.There are many papers which use of elliptic curves, or abelian varieties in general, to prove various undecidability results, including [Den80,Vid00b,Poo02,MRS22].We refer to [Shl09] for additional background.As an example which is relevant to the consideration of totally imaginary extensions of totally real fields, consider the following theorem of Shlapentokh.
Theorem 1.5 (Main Theorem B, [Shl09]).Let K be a totally real algebraic extension of Q which has a totally real extension of degree 2, and let K ′ be a finite extension of K such that there exists an elliptic curve E defined over K ′ with E(K ′ ) finitely generated and of positive rank.If L is a quadratic totally imaginary extension of K, then Z is existentially definable in the ring of integers O K and Hilbert's Tenth Problem is unsolvable over O K .
Although Theorems 1.4 and 1.5 are similar insofar as they both apply in the context of totally imaginary quadratic extensions of totally real subfields of Q, it is instructive to also note the differences and complementary strengths.Heuristically, it is easiest to apply the elliptic curve-based methods to relatively "small" algebraic extensions of Q over which it is easy to find elliptic curves with a finitely generated group of rational points.In contrast, the unit group-based methods work best for "bigger" fields in which there is an abundance of units available for manipulation.We can make this more precise with a couple of examples.
The prototypical example of an infinite algebraic extension K ⊇ Q which satisfies the hypotheses of Theorem 1.5 is a Z p -extension of Q; see [Shl09, §10].These Z p -extensions are difficult to handle with the unit group-based methods because there is at most one subextension of K of any given degree, hence a paucity of units.We also note that Theorem 1.5 proves that Hilbert's Tenth Problem is unsolvable, rather than only showing that the first-order theory is undecidable, and it also applies when [K : Q] < ∞.
On the other hand, when deploying elliptic curves, we emphasize that it is necessary for the group of rational points to be a finitely generated group.This is a core requirement of the proof method, and the restriction would remain for any straightforward variant or generalization which uses abelian varieties instead of elliptic curves, such as [MRS22, Theorem 1.1].Therefore, the following theorem of Fehm and Petersen shows that this general method is not useable when the field K is large, in the sense of Pop [Pop96].
Theorem 1.6 (Theorem 1.2, [FP10]).If L ⊆ Q is a large field and A/L is an abelian variety, then A(L) has infinite rank.
The first proof of this theorem in the case of elliptic curves is credited to Tamagawa; see Kobayashi [Kob06, Proposition 1].We refer to [FJ74, LR08, MR18, Pet06] for some additional results on finitely and non-finitely generated groups of rational points.
Because any algebraic extension of a large field is itself large, Theorem 1.6 shows that if K ⊆ Q is a large totally real field, then abelian variety-based methods such as Theorem 1.5 cannot handle K or its quadratic totally imaginary extensions.However, it is clear that the unit group-based methods presented in this paper can work for large fields because the field Q tr is large.It is also conjectured that Q ab is a large field (see [BSF13,§3] for background and an overview), and it is easy to construct extensions L ⊇ Q ab which are covered by Theorem 1.4.It would be interesting to determine whether or not the methods of this paper can be refined to prove the undecidability of the ring of integers of Q ab itself.

Non-maximal orders and units
As indicated in Section 1.3, given a totally real field K and a totally imaginary quadratic extension L, we want a subring R ⊆ O L which does not contain any nontrivial roots of unity.We start by defining the desired ring, then proceed to analyze its group of units.

2.1.
A useful non-maximal subring.Definition 2.1.Given an integer m ≥ 1 and a field L ⊆ Q, let R m,L be the subset of O L defined by the positive existential formula ϕ m (x), given as follows: The following properties of R m,L are immediate from the definition.
Lemma 2.2.Let L ⊆ Q be a field and let m ≥ 1 be an integer.
Proof.By the definition of R m,L and the formula ϕ m (x) above, the first claim is obvious.Moreover, R m,L is the preimage of Z/mZ under the natural surjective map O L → O L /mO L .Therefore, R m,L is a subring of O L and non-maximal precisely when Z/mZ = O L /mO L , i.e., when m ≥ 2 and L = Q.
Before we analyze the unit group R × m,L , we recall some general elementary facts.In essence, this theorem clarifies that any ring containing an algebraic unit u also contains u −1 , and computes the rank of the unit groups of any ring of algebraic integers.Given a set S ⊆ Q, let µ(S) denote the set of roots of unity contained in S.
Theorem 2.3.Let L ⊆ Q be a number field with r real and 2s imaginary embeddings.
Proof.An algebraic unit u has minimal polynomial m(x) = x n + c n−1 x n−1 • • • + c 1 x ± 1 with integer coefficients.Evaluating this polynomial at u, along with rearranging terms, shows that u( L then u reduces modulo N to an element of (O L /NO L ) × , and L and we are done.As an application, this shows the structure of the unit group of the ring R m,L .
Proposition 2.4.Let m ≥ 2 be an integer.If L is a number field with r real embeddings and 2s imaginary embeddings, then In particular, the only roots of unity contained in R m,L are trivial.
Proof.By Theorem 2.3, we only need to show that R m,L has no roots of unity other than ±1.
m,L is a nontrivial root of unity, then L contains the nontrivial cyclotomic subfield is the maximal order of L 0 , while R m,L 0 is a non-maximal order, which is a contradiction.

2.2.
Totally imaginary extensions of totally real fields.We now restrict our attention to the main focus of this paper: totally imaginary quadratic extensions of totally real fields.The following is a generalization of [Was97, Theorem 4.12] to the case of non-maximal orders.Given a field L and a subring O ⊆ L, we write µ(O) for the set of roots of unity in O.
Theorem 2.5.Let K be a totally real field, let L be a totally imaginary quadratic extension, and let O ⊆ L be an order which is stable under complex conjugation.If u ∈ O × , then its complex conjugate is u = ζu where ζ ∈ µ(O).In particular, writing We now wish to apply this theorem to the non-maximal subrings defined in the previous section.In our context, this theorem is helpful because we define subrings with no nontrivial roots of unity.Before we move on, we note that the application of Theorem 2.5 in the case when O = O L is a central ingredient in [MRUV20,Spr20].The following corollary was also inspired by an analogous fact for maximal orders [Was97, Proof of Proposition 1.5].
Remark 2.6.It is easy to show, e.g., using Magma [BCP97], that it is possible to have R × 2,L ⊆ K in Corollary 2.7.Indeed, this occurs for the cyclotomic field L = Q(ζ 15 ).Thus, squaring the group of units is necessary in the statement of the corollary when m = 2.
Corollary 2.7.Let K be a totally real field with totally imaginary quadratic extension L.
Clearly R m,L is stable under complex conjugation by definition.Combining Proposition 2.4 and Theorem 2.5, either u = u or u = −u.To finish the proof, assume m ≥ 3 and suppose that u = −u.Writing u = mb + j for b ∈ O L and j ∈ Z we have However, this implies that m divides 2 because u is a unit, and this is impossible because m ≥ 3. We conclude that u = u is totally real.
We now show how to deploy the unit groups described in the previous section.To do this, we explicitly write certain totally real algebraic integers as the sum of units.
It is easy to compute directly that u 1 and u 2 are inverses of each other, and hence units from R 2,K 1 : for some u, v ∈ R × 2,K 1 .Now we are ready to define the desired totally real subsets X within the totally imaginary fields.This can be seen as a generalization of [MRUV20, Lemma 7] and [Spr20, Lemma 2.7].
Theorem 3.3.Let K 0 be a totally real field and let K 1 be the maximal totally real subfield of K (2) 0 .If K ⊇ K 1 is a totally real field, then for every totally imaginary quadratic extension L of K, there is an existentially definable subset X ⊆ O L satisfying Proof.We write We have X 0 ⊆ O K by Corollary 2.7, and the set contains all elements d ∈ O K 0 whose conjugates are all outside (−1, 1) by Lemma 3.2.We repeat the same trick from before to finish the proof: If d ∈ O K 0 , then (d − 1) 2 and (d + 1) 2 are totally nonnegative.In particular, the elements (d −1) 2 + 1 and (d + 1) 2 + 1 are contained in X 1 because their conjugates lie inside [1, ∞), and outside (−1, 1).Thus, we may write 4d as the difference of these two elements of X 1 , namely 4d = (d + 1) Because the ring R 2,L is existentially definable in O L , this completes the proof.
Remark 3.4.The definition of the subset X in Theorem 3.3 is uniform in the sense that the formula defining X does not depend on the choices of K 0 , K 1 , K, or L.
By applying the previous theorem in the case where K 0 = K, we find that the ring of integers O K itself is existentially definable in O L , as written in the following theorem.Recall that K is closed under square roots if α ∈ K whenever α ∈ Q is totally real and α 2 ∈ K. Theorem 3.5.If K is a totally real field which is closed under square roots and L is any quadratic totally imaginary extension of K, then O K is existentially definable in O L .
Since Q tr is closed under square roots, we obtain the definability portion of Theorem 1.1.
Corollary 3.6.Z tr is existentially definable in the ring of integers of Q tr (i).
Throughout this section, we have exploited the fact that, within the fields K that we consider, many algebraic integers can be written as the sum of a bounded number of units.For contrast, note that Frey and Jarden showed the following when F is a number field: 4.2.Undecidability for totally imaginary O L .We are now ready to deploy a modified version of the JR-method for totally imaginary fields; compare with [MRUV20,Spr20].
Theorem 4.4.Let K 0 be a totally real field for which JR(O K 0 ) is either an attained minimum or ∞, and let K 1 be the maximal totally real subfield of K (2) 0 .If K ⊇ K 1 is any totally real field with JR(O K ) = JR(O K 0 ), and L is any totally imaginary quadratic extension of K, then the first order theory of O L is undecidable.Proof.By Theorem 3.3, there is an existentially definable subset X ⊆ O L satisfying Therefore, the parametrized formula φ X (x; a, b) defined by ∃y 0 , . . ., y 8 ∈ X[bxy 2 0 = 0 ∧ bxy 2 0 = a ∧ xy 2 0 = y 2 1 + • • • + y 2 4 ∧ (a − bxy 2 0 ) = y 2 5 + • • • + y 2 8 ] defines sets which satisfy the containments The containments follow from Theorem 4.3.Because JR(O K ) = JR(O K 0 ) is either infinite or an attained minimum, the sets on the lefthand and righthand sides above are finite sets of arbitrarily large size as a b ranges over positive rational numbers approaching JR(O K ) from the left.Therefore, by Lemma 4.1, we have proved first-order undecidability.
There are two extremes for the JR-number of a ring of totally real algebraic integers, namely 4 and ∞.For example, the fields Applying Theorem 4.4 in this case leads to a recovery of some of the results that appeared in [MRUV20,Spr20].At the other extreme, we write the following.
Theorem 4.5.Let S be any infinite set of positive integers, let K 0 = Q({ζ n + ζ n : n ∈ S}), and let K 1 be the maximal totally real subfield of K (2) 0 .If K is any totally real extension of K 1 and L is any totally imaginary quadratic extension of K, then the first order theory of O L is undecidable.
Proof.Under the given hypotheses, JR(O K 0 ) = 4 is the smallest possible JR-number and is realized as a minimum.Indeed, we have 0 ≪ ζ n + ζ n + 2 ≪ 4 for all n ∈ S, and these are the only totally real elements of Q with this property; see Kronecker [Kro57] and also [Rob64].Therefore, JR(O K 0 ) = JR(O K ) is clear and Theorem 4.4 applies.
In recent years, examples of totally real fields K have been discovered for which JR(O K ) is neither 4 nor ∞, i.e., JR(O K ) is non-extremal; see [Cas18,CVV20,GR17,VV15a].A better understanding of the behavior of JR-numbers under field extension would be required before applying Theorem 4.4 to fields K 0 for which JR(O K 0 ) ∈ (4, ∞) is non-extremal.

Lemma 3. 2 .
Let K 0 be a totally real field.If d ∈ O K 0 and all conjugates of d are outside the open interval (−1, 1) ⊆ R, then )) is totally real.Proof.It is easy to see that the values of the function f (x) = x 2 + x are negative precisely when x ∈ (−1, 0), so the element d 2 + d is totally nonnegative if and only if all conjugates of d lie outside of (−1, 0).Similarly, (d − 1) 2 + (d − 1) is totally nonnegative if and only if all conjugates of d lie outside of (0, 1).Therefore, if the conjugates of d lie outside of (−1, 1), then elements d 2 + d and (d − 1) 2 + (d − 1) are both totally nonnegative, so their square