Geodesic
Euclidean Rings of Algebraic Integers
Canadian journal of mathematics, Tome 56 (2004) no. 1, pp. 71-76

Voir la notice de l'article provenant de la source Cambridge University Press

Let $K$ be a finite Galois extension of the field of rational numbers with unit rank greater than 3. We prove that the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally.
DOI : 10.4153/CJM-2004-004-5
Mots-clés : 11R04, 11R27, 11R32, 11R42, 11N36 
Harper, Malcolm; Murty, M. Ram. Euclidean Rings of Algebraic Integers. Canadian journal of mathematics, Tome 56 (2004) no. 1, pp. 71-76. doi: 10.4153/CJM-2004-004-5
@article{10_4153_CJM_2004_004_5,
     author = {Harper, Malcolm and Murty, M. Ram},
     title = {Euclidean {Rings} of {Algebraic} {Integers}},
     journal = {Canadian journal of mathematics},
     pages = {71--76},
     year = {2004},
     volume = {56},
     number = {1},
     doi = {10.4153/CJM-2004-004-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2004-004-5/}
}
TY  - JOUR
AU  - Harper, Malcolm
AU  - Murty, M. Ram
TI  - Euclidean Rings of Algebraic Integers
JO  - Canadian journal of mathematics
PY  - 2004
SP  - 71
EP  - 76
VL  - 56
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2004-004-5/
DO  - 10.4153/CJM-2004-004-5
ID  - 10_4153_CJM_2004_004_5
ER  - 
%0 Journal Article
%A Harper, Malcolm
%A Murty, M. Ram
%T Euclidean Rings of Algebraic Integers
%J Canadian journal of mathematics
%D 2004
%P 71-76
%V 56
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2004-004-5/
%R 10.4153/CJM-2004-004-5
%F 10_4153_CJM_2004_004_5

[BFI] Bombieri, E., Friedlander, J. and Iwaniec, H., Primes in arithmetic progressions to large moduli. Acta Math. 156(1986), 203–251. Google Scholar

[CM] Clark, D. and Murty, M. Ram, The Euclidean algorithm for Galois extensions of ℚ . J. Reine Angew. Math. 459(1995), 151–162. Google Scholar

[GM] Gupta, R. and Murty, M. Ram, A remark on Artin's conjecture. Invent. Math. 78(1984), 127–130. Google Scholar

[GMM] Gupta, R., Murty, M. Ram and Murty, V. Kumar, The Euclidean algorithm for S-integers. In: Number Theory (eds. Kisilevsky, H. and Labute, J.), CMS Conf. Proc. 7(1987), 189–201. Google Scholar

[H] Harper, M., ℤ[] is Euclidean. Canad. J. Math. 56(2004), 55–70. Google Scholar

[L] Lenstra, H., Euclidean number fields, I. Math. Intelligencer 2(1979), 6–15. Google Scholar

[M] Motzkin, Th., The Euclidean algorithm. Bull. Amer. Math. Soc. 55(1949), 1142–1146. Google Scholar

[KM] Murty, V. Kumar, Bounded and Finite Generation of Arithmetic Groups. In: Number Theory (ed. Dilcher, K.), CMS Conf. Proc. 15(1995), 249–261. Google Scholar

[MM] Murty, M. Ram and Murty, V. Kumar, A variant of the Bombieri-Vinogradov theorem. In: Number Theory (eds. Kisilevsky, H. and Labute, J.), CMS Conf. Proc. 7(1987), 243–272. Google Scholar

[S] Samuel, P., About Euclidean Rings. J. Algebra 19(1971), 282–301. Google Scholar

[W] Weinberger, P., On Euclidean rings of algebraic integers. In: Analytic Number Theory (St. Louis, 1972), Proc. Sympos. Pure Math. 24(1973), 321–332. Google Scholar

Cité par Sources :