On the fundamental units of a totally real cubic order generated by a unit
Author:
Stéphane R. Louboutin
Journal:
Proc. Amer. Math. Soc. 140 (2012), 429436
MSC (2010):
Primary 11R16, 11R27
Published electronically:
June 9, 2011
MathSciNet review:
2846312
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We give a new and short proof of J. Beers, D. Henshaw, C. McCall, S. Mulay and M. Spindler following a recent result: if is a totally real cubic algebraic unit, then there exists a unit such that is a system of fundamental units of the group of the units of the cubic order , except for an infinite family for which is a square in and one sporadic exception. Not only is our proof shorter, but it enables us to prove a new result: if the conjugates and of are in , then the subgroup generated by and is of bounded index in , and if and if and are of opposite sign, then is a system of fundamental units of .
 [BHMMS]
J.
Beers, D.
Henshaw, C.
K. McCall, S.
B. Mulay, and M.
Spindler, Fundamentality of a cubic unit
𝑢 for ℤ[𝕦], Math.
Comp. 80 (2011), no. 273, 563–578. MR 2728994
(2011k:11151), 10.1090/S00255718201002383X
 [Cus]
T.
W. Cusick, Lower bounds for regulators, Number theory,
Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math.,
vol. 1068, Springer, Berlin, 1984, pp. 63–73. MR 756083
(85k:11052), 10.1007/BFb0099441
 [Lou02]
Stéphane
Louboutin, The exponent three class group problem
for some real cyclic cubic number fields, Proc.
Amer. Math. Soc. 130 (2002), no. 2, 353–361. MR 1862112
(2002h:11106), 10.1090/S0002993901061688
 [Lou06]
Stéphane
R. Louboutin, The classnumber one problem for some real cubic
number fields with negative discriminants, J. Number Theory
121 (2006), no. 1, 30–39. MR 2268753
(2007k:11189), 10.1016/j.jnt.2006.01.008
 [Lou08]
Stéphane
R. Louboutin, The fundamental unit of some quadratic, cubic or
quartic orders, J. Ramanujan Math. Soc. 23 (2008),
no. 2, 191–210. MR 2432797
(2009h:11175)
 [Lou10]
Stéphane
Louboutin, On some cubic or quartic algebraic units, J. Number
Theory 130 (2010), no. 4, 956–960. MR 2600414
(2011b:11156), 10.1016/j.jnt.2009.09.002
 [Nag]
Trygve
Nagell, Zur Theorie der kubischen Irrationalitäten, Acta
Math. 55 (1930), no. 1, 33–65 (German). MR
1555314, 10.1007/BF02546509
 [PL]
S.M.
Park and G.N.
Lee, The class number one problem for some totally complex quartic
number fields, J. Number Theory 129 (2009),
no. 6, 1338–1349. MR 2521477
(2010d:11126), 10.1016/j.jnt.2009.01.022
 [Tho]
Emery
Thomas, Fundamental units for orders in certain cubic number
fields, J. Reine Angew. Math. 310 (1979),
33–55. MR
546663 (81b:12009)
 [BHMMS]
 J. Beers, D. Henshaw, C. McCall, S. Mulay and M. Spindler,
Fundamentality of a cubic unit for . Math. Comp. 80 (2011), 563578. MR 2728994
 [Cus]
 T. W. Cusik.
Lower bounds for regulators. Lecture Notes in Math. 1068 (1984), 6373. MR 756083 (85k:11052)
 [Lou02]
 S. Louboutin.
The exponent three class group problem for some real cyclic cubic number fields. Proc. Amer. Math. Soc. 130 (2002), 353361. MR 1862112 (2002h:11106)
 [Lou06]
 S. Louboutin.
The classnumber one problem for some real cubic number fields with negative discriminants. J. Number Theory 121 (2006), 3039. MR 2268753 (2007k:11189)
 [Lou08]
 S. Louboutin,
The fundamental unit of some quadratic, cubic or quartic orders. J. Ramanujan Math. Soc. 23, No.2 (2008), 191210. MR 2432797 (2009h:11175)
 [Lou10]
 S. Louboutin,
On some cubic or quartic algebraic units. J. Number Theory 130 (2010), 956960. MR 2600414 (2011b:11156)
 [Nag]
 T. Nagell.
Zur Theorie der kubischen Irrationalitäten. Acta Math. 55 (1930), 3365. MR 1555314
 [PL]
 S.M. Park and G.N. Lee.
The class number one problem for some totally complex quartic number fields. J. Number Theory 129 (2009), 13381349. MR 2521477 (2010d:11126)
 [Tho]
 E. Thomas.
Fundamental units for orders in certain cubic number fields. J. Reine Angew. Math. 310 (1979), 3355. MR 546663 (81b:12009)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
11R16,
11R27
Retrieve articles in all journals
with MSC (2010):
11R16,
11R27
Additional Information
Stéphane R. Louboutin
Affiliation:
Institut de Mathématiques de Luminy, UMR 6206, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
Email:
stephane.louboutin@univmed.fr
DOI:
http://dx.doi.org/10.1090/S000299392011109249
Keywords:
Fundamental units,
cubic orders
Received by editor(s):
June 15, 2010
Received by editor(s) in revised form:
September 21, 2010, and November 24, 2010
Published electronically:
June 9, 2011
Dedicated:
Dedicated to Florence F.
Communicated by:
Matthew A. Papanikolas
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
