Undecidability, unit groups, and some totally imaginary infinite extensions of ℚ

Springer, Caleb

doi:10.1090/proc/15153

Undecidability, unit groups, and some totally imaginary infinite extensions of $\mathbb {Q}$
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Proc. Amer. Math. Soc. 148 (2020), 4705-4715 Request permission

Abstract:

We produce new examples of totally imaginary infinite extensions of $\mathbb {Q}$ which have undecidable first-order theory by generalizing the methods used by Martínez-Ranero, Utreras, and Videla for $\mathbb {Q}^{(2)}$. In particular, we use parametrized families of polynomials whose roots are totally real units to apply methods originally developed to prove the undecidability of totally real fields. This proves the undecidability of $\mathbb {Q}^{(d)}_{ab}$ for all $d \geq 2$.

References

Enrico Bombieri and Umberto Zannier, A note on heights in certain infinite extensions of $\Bbb Q$, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 12 (2001), 5–14 (2002) (English, with English and Italian summaries). MR 1898444
M. Castillo Fernández, On the Julia Robinson number of rings of totally real algebraic integers in some towers of nested square roots, PhD thesis, Universidad de Concepción, 2018. http://repositorio.udec.cl/handle/11594/3003.
M. Castillo Fernández, X. Vidaux, and C. R. Videla, Julia Robinson numbers and arithmetical dynamic of quadratic polynomials, arXiv e-prints, 2017.
Sara Checcoli and Martin Widmer, On the Northcott property and other properties related to polynomial mappings, Math. Proc. Cambridge Philos. Soc. 155 (2013), no. 1, 1–12. MR 3065255, DOI 10.1017/S0305004113000042
Martin Davis, Hilary Putnam, and Julia Robinson, The decision problem for exponential diophantine equations, Ann. of Math. (2) 74 (1961), 425–436. MR 133227, DOI 10.2307/1970289
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
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
Pierre Gillibert and Gabriele Ranieri, Julia Robinson numbers, Int. J. Number Theory 15 (2019), no. 8, 1565–1599. MR 3994148, DOI 10.1142/S1793042119500908
M.-N. Gras, Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré 4 de $\mathbb {Q}$, Publications Mathématiques de Besançon (1977/78), 1–79.
Marie-Nicole Gras, Special units in real cyclic sextic fields, Math. Comp. 48 (1987), no. 177, 179–182. MR 866107, DOI 10.1090/S0025-5718-1987-0866107-1
E. Kamke, Verallgemeinerungen des Waring-Hilbertschen Satzes, Math. Ann. 83 (1921), no. 1-2, 85–112 (German). MR 1512001, DOI 10.1007/BF01464230
Yasuhiro Kishi, A family of cyclic cubic polynomials whose roots are systems of fundamental units, J. Number Theory 102 (2003), no. 1, 90–106. MR 1994474, DOI 10.1016/S0022-314X(03)00085-4
Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
Carlos Martínez-Ranero, Javier Utreras, and Carlos R. Videla, Undecidability of $\Bbb Q^{(2)}$, Proc. Amer. Math. Soc. 148 (2020), no. 3, 961–964. MR 4055926, DOI 10.1090/proc/14849
Ju. V. Matijasevič, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279–282 (Russian). MR 0258744
Julia Robinson, Definability and decision problems in arithmetic, J. Symbolic Logic 14 (1949), 98–114. MR 31446, DOI 10.2307/2266510
Julia Robinson, The undecidability of algebraic rings and fields, Proc. Amer. Math. Soc. 10 (1959), 950–957. MR 112842, DOI 10.1090/S0002-9939-1959-0112842-7
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
R. S. Rumely, Undecidability and definability for the theory of global fields, Trans. Amer. Math. Soc. 262 (1980), no. 1, 195–217. MR 583852, DOI 10.1090/S0002-9947-1980-0583852-6
René Schoof and Lawrence C. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), no. 182, 543–556. MR 929552, DOI 10.1090/S0025-5718-1988-0929552-2
Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 352049, DOI 10.1090/S0025-5718-1974-0352049-8
Alexandra Shlapentokh, First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers, Israel J. Math. 226 (2018), no. 2, 579–633. MR 3819703, DOI 10.1007/s11856-018-1708-y
Carl Siegel, Darstellung total positiver Zahlen durch Quadrate, Math. Z. 11 (1921), no. 3-4, 246–275 (German). MR 1544496, DOI 10.1007/BF01203627
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
Xavier Vidaux and Carlos R. Videla, Definability of the natural numbers in totally real towers of nested square roots, Proc. Amer. Math. Soc. 143 (2015), no. 10, 4463–4477. MR 3373945, DOI 10.1090/S0002-9939-2015-12592-0
Xavier Vidaux and Carlos R. Videla, A note on the Northcott property and undecidability, Bull. Lond. Math. Soc. 48 (2016), no. 1, 58–62. MR 3455748, DOI 10.1112/blms/bdv089
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
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
Martin Widmer, On certain infinite extensions of the rationals with Northcott property, Monatsh. Math. 162 (2011), no. 3, 341–353. MR 2775852, DOI 10.1007/s00605-009-0162-7
Robert S. Wolf, A tour through mathematical logic, Carus Mathematical Monographs, vol. 30, Mathematical Association of America, Washington, DC, 2005. MR 2112516