## Recognizing units in number fields

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- by Guoqiang Ge PDF
- Math. Comp.
**63**(1994), 377-387 Request permission

## Abstract:

We present a deterministic polynomial-time algorithm that decides whether a power product $\prod \nolimits _{i = 1}^k {\gamma _i^{{n_i}}}$ is a unit in the ring of integers of*K*, where

*K*is a number field, ${\gamma _i}$ are nonzero elements of

*K*and ${n_i}$ are rational integers. The main algorithm is based on the factor refinement method for ideals, which might be of independent interest.

## References

- M. F. Atiyah and I. G. Macdonald,
*Introduction to commutative algebra*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR**0242802** - Eric Bach, James Driscoll, and Jeffrey Shallit,
*Factor refinement*, J. Algorithms**15**(1993), no. 2, 199–222. MR**1231441**, DOI 10.1006/jagm.1993.1038 - A. I. Borevich and I. R. Shafarevich,
*Number theory*, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR**0195803**
J. A. Buchmann and H. W. Lenstra, Jr., - J. W. S. Cassels and A. Fröhlich (eds.),
*Algebraic number theory*, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR**0215665** - A. L. Chistov,
*The complexity of the construction of the ring of integers of a global field*, Dokl. Akad. Nauk SSSR**306**(1989), no. 5, 1063–1067 (Russian); English transl., Soviet Math. Dokl.**39**(1989), no. 3, 597–600. MR**1014763** - James L. Hafner and Kevin S. McCurley,
*Asymptotically fast triangularization of matrices over rings*, SIAM J. Comput.**20**(1991), no. 6, 1068–1083. MR**1135749**, DOI 10.1137/0220067 - Serge Lang,
*Algebraic number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR**1282723**, DOI 10.1007/978-1-4612-0853-2 - H. W. Lenstra Jr.,
*Algorithms in algebraic number theory*, Bull. Amer. Math. Soc. (N.S.)**26**(1992), no. 2, 211–244. MR**1129315**, DOI 10.1090/S0273-0979-1992-00284-7 - Hideyuki Matsumura,
*Commutative ring theory*, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR**879273** - Alexander Schrijver,
*Theory of linear and integer programming*, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Ltd., Chichester, 1986. A Wiley-Interscience Publication. MR**874114** - Edwin Weiss,
*Algebraic number theory*, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR**0159805**

*Approximating rings of integers in number fields*, in preparation.

## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Math. Comp.
**63**(1994), 377-387 - MSC: Primary 11Y40; Secondary 11R27
- DOI: https://doi.org/10.1090/S0025-5718-1994-1242057-X
- MathSciNet review: 1242057