Nontrivial Galois module structure of cyclotomic fields

Authors:
Marc Conrad and Daniel R. Replogle

Journal:
Math. Comp. **72** (2003), 891-899

MSC (2000):
Primary 11R33, 11R29; Secondary 11R27, 11R18

DOI:
https://doi.org/10.1090/S0025-5718-02-01457-6

Published electronically:
June 4, 2002

MathSciNet review:
1954973

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We say a tame Galois field extension with Galois group has trivial Galois module structure if the rings of integers have the property that is a free -module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes so that for each there is a tame Galois field extension of degree so that has nontrivial Galois module structure. However, the proof does not directly yield specific primes for a given algebraic number field For any cyclotomic field we find an explicit so that there is a tame degree extension with nontrivial Galois module structure.

**[1]**S-P Chan and C-H Lim,*Relative Galois module structure of rings of integers of cyclotomic fields*, J. Reine Angew. Math.**434**(1993), 205-230. MR**93i:11127****[2]**M. Conrad,*Construction of bases for the group of cyclotomic units*, J. Num. Theory**81**(2000), 1-15. MR**2001f:11182****[3]**A. Fröhlich,*Galois Module Structure of Algebraic Integers*, Springer-Verlag, Berlin, 1983. MR**85h:11067****[4]**C. Greither, D. R. Replogle, K. Rubin, and A. Srivastav,*Swan Modules and Hilbert-Speiser number fields*, J. Num. Theory**79**(1999), 164-173. MR**2000m:11111****[5]**T. Kohl and D. R. Replogle,*Computation of several Cyclotomic Swan Subgroups*, Math. Comp.,**71**(2002), 343-348.**[6]**H. B. Mann,*On integral bases*, Proc. Amer. Soc. Math.**9**(1958), 162-172. MR**20:26****[7]**L. R. McCulloh,*Galois Module Structure of Elementary Abelian Extensions*, J. Alg.**82**(1983), 102-134. MR**85d:11093****[8]**L. R. McCulloh,*Galois Module Structure of Abelian Extensions*, J. Reine Angew. Math.**375/376**(1987), 259-306. MR**88k:11080****[9]**I. Reiner and S. Ullom,*A Mayer-Vietoris sequence for class groups*, J. Alg.**31**(1974), 305-342. MR**50:2321****[10]**D. R. Replogle,*Swan Modules and Realisable Classes for Kummer Extensions of Prime Degree*, J. Alg.**212**(1999), 482-494. MR**2000a:11161****[11]**D. R. Replogle,*Cyclotomic Swan subgroups and irregular indices*, Rocky Mountain Journal Math.**31**(Summer 2001), 611-618.**[12]**D. R. Replogle and R. G. Underwood,*Nontrivial tame extensions over Hopf orders*, Acta Arithmetica (to appear).**[13]**The SIMATH group/H. G. Zimmer,*SIMATH: A computer algebra system for algebraic number theory*, www.simath.info.**[14]**W. Sinnott,*On the Stickelberger ideal and the circular units of a cyclotomic field*, Annals Math.**108**(1978), 107-134. MR**58:5585****[15]**S. V. Ullom,*Nontrivial lower bounds for class groups of integral group rings*, Illinois Journal Mathematics**20**(1976), 361-371. MR**52:14024****[16]**L. Washington,*Introduction to Cyclotomic Fields*, Graduate Texts in Mathematics 83, Springer-Verlag, New York, 1982. MR**85g:11001**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
11R33,
11R29,
11R27,
11R18

Retrieve articles in all journals with MSC (2000): 11R33, 11R29, 11R27, 11R18

Additional Information

**Marc Conrad**

Affiliation:
Faculty of Technology, Southampton Institute, East Park Terrace, Southampton, S014 0YN Great Britain

Email:
marc@pension-perisic.de

**Daniel R. Replogle**

Affiliation:
Department of Mathematics and Computer Science, College of Saint Elizabeth, 2 Convent Road, Morristown, New Jersey 07960

Email:
dreplogle@cse.edu

DOI:
https://doi.org/10.1090/S0025-5718-02-01457-6

Keywords:
Swan subgroups,
cyclotomic units,
Galois module structure,
tame extension,
normal integral basis

Received by editor(s):
November 6, 2000

Received by editor(s) in revised form:
July 15, 2001

Published electronically:
June 4, 2002

Article copyright:
© Copyright 2002
American Mathematical Society