Definability of the natural numbers in totally real towers of nested square roots
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- by Xavier Vidaux and Carlos R. Videla PDF
- Proc. Amer. Math. Soc. 143 (2015), 4463-4477 Request permission
Abstract:
For the ring of integers $\mathcal {O}$ of a totally real algebraic field, Julia Robinson defines a set $A(\mathcal {O})$ such that either $A(\mathcal {O})=\{+\infty \}$ or it is an interval in $\mathbb {R}$. She then proves that if this set has a minimum, then the natural numbers can be defined in $\mathcal {O}$, and hence $\mathcal {O}$ has undecidable first-order theory. All known examples are such that $A(\mathcal {O})$ has a minimum which is either $4$ or $+\infty$. In this work, we construct two infinite families of subrings of such rings for which $\inf (A(\mathcal {O}))$ is strictly between $4$ and $+\infty$. In one family, the infimum is a minimum, whereas in the other family it is not, but we can still define the natural numbers in this case.References
- Lou van den Dries, Elimination theory for the ring of algebraic integers, J. Reine Angew. Math. 388 (1988), 189–205. MR 944190, DOI 10.1515/crll.1988.388.189
- Kenji Fukuzaki, Definability of the ring of integers in some infinite algebraic extensions of the rationals, MLQ Math. Log. Q. 58 (2012), no. 4-5, 317–332. MR 2965419, DOI 10.1002/malq.201110020
- Michael D. Fried, Dan Haran, and Helmut Völklein, Real Hilbertianity and the field of totally real numbers, Arithmetic geometry (Tempe, AZ, 1993) Contemp. Math., vol. 174, Amer. Math. Soc., Providence, RI, 1994, pp. 1–34. MR 1299732, DOI 10.1090/conm/174/01849
- Moshe Jarden and Carlos R. Videla, Undecidability of families of rings of totally real integers, Int. J. Number Theory 4 (2008), no. 5, 835–850. MR 2458847, DOI 10.1142/S1793042108001705
- J. Koenigsmann, Undecidability in number theory, Model Theory in Algebra, Analysis and Arithmetic, Lecture Notes in Mathematics, (2014) 159-195.
- Julia Robinson, On the decision problem for algebraic rings, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 297–304. MR 0146083
- Robert S. Rumely, Arithmetic over the ring of all algebraic integers, J. Reine Angew. Math. 368 (1986), 127–133. MR 850618, DOI 10.1515/crll.1986.368.127
- Alexandra Shlapentokh, Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of $\textbf {Q}$, Ann. Pure Appl. Logic 68 (1994), no. 3, 299–325. MR 1289287, DOI 10.1016/0168-0072(94)90024-8
- Alexandra Shlapentokh, On Diophantine definability and decidability in some infinite totally real extensions of $\Bbb Q$, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3189–3207. MR 2052946, DOI 10.1090/S0002-9947-03-03343-9
- Alexandra Shlapentokh, Rings of algebraic numbers in infinite extensions of $\Bbb Q$ and elliptic curves retaining their rank, Arch. Math. Logic 48 (2009), no. 1, 77–114. MR 2480937, DOI 10.1007/s00153-008-0118-y
- Alexandra Shlapentokh, First order decidability and definability of integers in infinite algebraic extensions of rational numbers, arXiv:1307.0743v2.
- Carlos R. Videla, Definability of the ring of integers in pro-$p$ Galois extensions of number fields, Israel J. Math. 118 (2000), 1–14. MR 1776073, DOI 10.1007/BF02803513
- Carlos R. Videla, The undecidability of cyclotomic towers, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3671–3674. MR 1694882, DOI 10.1090/S0002-9939-00-05544-1
Additional Information
- Xavier Vidaux
- Affiliation: Departamento de Matemática, Universidad de Concepción, Casilla 160 C, Concepción, Chile
- Email: xvidaux@udec.cl
- Carlos R. Videla
- Affiliation: Department of Mathematics, Physics and Engineering, Mount Royal University, 4825 Mount Royal Gate SW, Calgary, Alberta, Canada T3E 6K6
- MR Author ID: 178355
- Email: cvidela@mtroyal.ca
- Received by editor(s): November 26, 2013
- Received by editor(s) in revised form: June 28, 2014
- Published electronically: March 18, 2015
- Additional Notes: Both authors have been supported by the first author’s Chilean research projects Fondecyt 1090233 and 1130134, and by Mount Royal University PD Funds.
- Communicated by: Mirna Džamonja
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 4463-4477
- MSC (2010): Primary 03B25; Secondary 11U05, 11R80
- DOI: https://doi.org/10.1090/S0002-9939-2015-12592-0
- MathSciNet review: 3373945