Algebra & Number Theory

On the relative Galois module structure of rings of integers in tame extensions

Adebisi Agboola and Leon R. McCulloh

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let F be a number field with ring of integers OF and let G be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group Cl(OFG) of OFG that involves applying the work of McCulloh in the context of relative algebraic K theory. For a large class of soluble groups G, including all groups of odd order, we show (subject to certain mild conditions) that the set of realisable classes is a subgroup of Cl(OFG). This may be viewed as being a partial analogue in the setting of Galois module theory of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups.

Article information

Source
Algebra Number Theory, Volume 12, Number 8 (2018), 1823-1886.

Dates
Received: 28 August 2015
Revised: 23 March 2018
Accepted: 2 July 2018
First available in Project Euclid: 21 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1545361461

Digital Object Identifier
doi:10.2140/ant.2018.12.1823

Mathematical Reviews number (MathSciNet)
MR3892966

Zentralblatt MATH identifier
06999396

Subjects
Primary: 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10]
Secondary: 11R32: Galois theory 11R65: Class groups and Picard groups of orders 19F99: None of the above, but in this section

Keywords
Galois module structure realisable classes rings of integers inverse Galois problem relative K-group

Citation

Agboola, Adebisi; McCulloh, Leon R. On the relative Galois module structure of rings of integers in tame extensions. Algebra Number Theory 12 (2018), no. 8, 1823--1886. doi:10.2140/ant.2018.12.1823. https://projecteuclid.org/euclid.ant/1545361461


Export citation

References

  • A. Agboola, “On counting rings of integers as Galois modules”, J. Reine Angew. Math. 663 (2012), 1–31.
  • A. Agboola and D. Burns, “On the Galois structure of equivariant line bundles on curves”, Amer. J. Math. 120:6 (1998), 1121–1163.
  • A. Agboola and D. Burns, “Grothendieck groups of bundles on varieties over finite fields”, $K$-Theory 23:3 (2001), 251–303.
  • A. Agboola and D. Burns, “On twisted forms and relative algebraic $K$-theory”, Proc. London Math. Soc. $(3)$ 92:1 (2006), 1–28.
  • M. I. Bueno, S. Furtado, J. Karkoska, K. Mayfield, R. Samalis, and A. Telatovich, “The kernel of the matrix $[ij\pmod n]$ when $n$ is prime”, Involve 9:2 (2016), 265–280.
  • D. M. Burton, Elementary number theory, McGraw-Hill, Boston, 2007.
  • N. P. Byott, “Tame realisable classes over Hopf orders”, J. Algebra 201:1 (1998), 284–316.
  • N. P. Byott and B. Sodaïgui, “Realizable Galois module classes for tetrahedral extensions”, Compos. Math. 141:3 (2005), 573–582.
  • N. P. Byott, C. Greither, and B. Sodaïgui, “Classes réalisables d'extensions non abéliennes”, J. Reine Angew. Math. 601 (2006), 1–27.
  • T. Chinburg, “Galois structure of de Rham cohomology of tame covers of schemes”, Ann. of Math. $(2)$ 139:2 (1994), 443–490.
  • C. W. Curtis and I. Reiner, Methods of representation theory, I: With applications to finite groups and orders, Wiley, New York, 1981.
  • C. W. Curtis and I. Reiner, Methods of representation theory, II: With applications to finite groups and orders, Wiley, New York, 1987.
  • M. Farhat and B. Sodaïgui, “Classes réalisables d'extensions non abéliennes de degré $p^3$”, J. Number Theory 152 (2015), 55–89.
  • W. Feit and J. G. Thompson, “Solvability of groups of odd order”, Pacific J. Math. 13 (1963), 775–1029.
  • A. Fröhlich, “Arithmetic and Galois module structure for tame extensions”, J. Reine Angew. Math. 286/287 (1976), 380–440.
  • A. Fröhlich, Galois module structure of algebraic integers, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 1, Springer, 1983.
  • A. Fröhlich, Classgroups and Hermitian modules, Progress in Mathematics 48, Birkhäuser, Boston, 1984.
  • D. Hilbert, The theory of algebraic number fields, Springer, 1998.
  • G. Malle, “On the distribution of Galois groups”, J. Number Theory 92:2 (2002), 315–329.
  • L. R. McCulloh, “Galois module structure of elementary abelian extensions”, J. Algebra 82:1 (1983), 102–134.
  • L. R. McCulloh, “Galois module structure of abelian extensions”, J. Reine Angew. Math. 375/376 (1987), 259–306.
  • L. R. McCulloh, “On realisable classes for non-abelian extensions”, lecture in Luminy, March 22 2011.
  • L. R. McCulloh, “From Galois module classes to Steinitz classes”, preprint, 2012.
  • J. Neukirch, “On solvable number fields”, Invent. Math. 53:2 (1979), 135–164.
  • J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 323, Springer, 2008.
  • J. J. Rotman, An introduction to the theory of groups, 4th ed., Graduate Texts in Mathematics 148, Springer, 1995.
  • J.-P. Serre, Galois cohomology, Springer, 1997.
  • J.-P. Serre, “On a theorem of Jordan”, Bull. Amer. Math. Soc. $($N.S.$)$ 40:4 (2003), 429–440.
  • I. R. Shafarevich, “Construction of fields of algebraic numbers with given solvable Galois group”, Izv. Akad. Nauk SSSR. Ser. Mat. 18 (1954), 525–578. In Russian; translated in Izv. Akad. Nauk SSSR. Ser. Mat. 18 (1954), 525–578.
  • A. Siviero, Class invariants for tame Galois algebras, Ph.D. thesis, Université de Bordeaux and Universiteit Leiden, 2013.
  • A. Siviero, “Realisable classes, Stickelberger subgroup and its behaviour under change of the base field”, pp. 69–92 in Publications mathématiques de Besançon: algèbre et théorie des nombres, 2015, Publ. Math. Besançon Algèbre Théorie Nr. 2015, Presses Univ., Franche-Comté, Besançon, 2016.
  • R. G. Swan, Algebraic $K$-theory, Lecture Notes in Mathematics 76, Springer, 1968.
  • R. G. Swan, $K$-theory of finite groups and orders, Lecture Notes in Mathematics 149, Springer, 1970.
  • M. Taylor, Classgroups of group rings, London Mathematical Society Lecture Note Series 91, Cambridge University Press, 1984.
  • C. Tsang, “On the Galois module structure of the square root of the inverse different in abelian extensions”, J. Number Theory 160 (2016), 759–804.
  • C. Tsang, “On the realizable classes of the square root of the inverse different in the unitary class group”, Int. J. Number Theory 13:4 (2017), 913–932.
  • D. J. Wright, “Distribution of discriminants of abelian extensions”, Proc. London Math. Soc. $(3)$ 58:1 (1989), 17–50.