Undecidable existential problems for addition and divisibility in algebraic number rings. II

Author:
Leonard Lipshitz

Journal:
Proc. Amer. Math. Soc. **64** (1977), 122-128

MSC:
Primary 02E10; Secondary 02G05, 10N10

MathSciNet review:
0536659

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that for all algebraic number rings, except imaginary quadratic ones, the problem of deciding existential formulas involving only addition and the divisibility predicate is equivalent to the full diophantine problem for these rings.

**[1]**A. I. Borevich and I. R. Shafarevich,*Number theory*, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. MR**0195803****[2]**R. D. Carmichael,*On the numerical factors of the arithmetic forms*, Ann. of Math. (2)**15**(1913).**[3]**L. Lipshitz,*The Diophantine problem for addition and divisibility*, Trans. Amer. Math. Soc.**235**(1978), 271–283. MR**0469886**, 10.1090/S0002-9947-1978-0469886-1**[4]**L. Lipshitz,*Undecidable existential problems for addition and divisibility in algebraic number rings*, Trans. Amer. Math. Soc.**241**(1978), 121–128. MR**0536658**, 10.1090/S0002-9947-1978-0536658-9**[5]**Hermann Weyl,*Algebraic Theory of Numbers*, Annals of Mathematics Studies, no. 1, Princeton University Press, Princeton, N. J., 1940. MR**0002354**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
02E10,
02G05,
10N10

Retrieve articles in all journals with MSC: 02E10, 02G05, 10N10

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1977-0536659-5

Article copyright:
© Copyright 1977
American Mathematical Society