New ramification breaks and additive Galois structure

Nigel P. Byott[1]; G. Griffith Elder[2]

  • [1] Department of Mathematical Sciences University of Exeter Exeter EX4 4QE United Kingdom
  • [2] Department of Mathematics University of Nebraska at Omaha Omaha, NE 68182-0243 U.S.A.

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 87-107
  • ISSN: 1246-7405

Abstract

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Which invariants of a Galois p -extension of local number fields L / K (residue field of char p , and Galois group G ) determine the structure of the ideals in L as modules over the group ring ℤ p [ G ] , ℤ p the p -adic integers? We consider this question within the context of elementary abelian extensions, though we also briefly consider cyclic extensions. For elementary abelian groups G , we propose and study a new group (within the group ring 𝔽 q [ G ] where 𝔽 q is the residue field) and its resulting ramification filtrations.

How to cite

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Byott, Nigel P., and Elder, G. Griffith. "New ramification breaks and additive Galois structure." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 87-107. <http://eudml.org/doc/249439>.

@article{Byott2005,
abstract = {Which invariants of a Galois $p$-extension of local number fields $L/K$ (residue field of char $p$, and Galois group $G$) determine the structure of the ideals in $L$ as modules over the group ring $\mathbb\{Z\}_p[G]$, $\mathbb\{Z\}_p$ the $p$-adic integers? We consider this question within the context of elementary abelian extensions, though we also briefly consider cyclic extensions. For elementary abelian groups $G$, we propose and study a new group (within the group ring $\mathbb\{F\}_q[G]$ where $\mathbb\{F\}_q$ is the residue field) and its resulting ramification filtrations.},
affiliation = {Department of Mathematical Sciences University of Exeter Exeter EX4 4QE United Kingdom; Department of Mathematics University of Nebraska at Omaha Omaha, NE 68182-0243 U.S.A.},
author = {Byott, Nigel P., Elder, G. Griffith},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Galois structure of ideals in biquadratic extensions; ramification filtrations},
language = {eng},
number = {1},
pages = {87-107},
publisher = {Université Bordeaux 1},
title = {New ramification breaks and additive Galois structure},
url = {http://eudml.org/doc/249439},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Byott, Nigel P.
AU - Elder, G. Griffith
TI - New ramification breaks and additive Galois structure
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 87
EP - 107
AB - Which invariants of a Galois $p$-extension of local number fields $L/K$ (residue field of char $p$, and Galois group $G$) determine the structure of the ideals in $L$ as modules over the group ring $\mathbb{Z}_p[G]$, $\mathbb{Z}_p$ the $p$-adic integers? We consider this question within the context of elementary abelian extensions, though we also briefly consider cyclic extensions. For elementary abelian groups $G$, we propose and study a new group (within the group ring $\mathbb{F}_q[G]$ where $\mathbb{F}_q$ is the residue field) and its resulting ramification filtrations.
LA - eng
KW - Galois structure of ideals in biquadratic extensions; ramification filtrations
UR - http://eudml.org/doc/249439
ER -

References

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  2. N. P. Byott, G. G. Elder, Biquadratic extensions with one break. Can. Math. Bull. 45 (2002), 168–179. Zbl1033.11054MR1904081
  3. G. G. Elder, Galois module structure of integers in wildly ramified cyclic extensions of degree p 2 . Ann. Inst. Fourier (Grenoble) 45 (1995), 625–647; errata ibid. 48 (1998), 609–610. Zbl0820.11070MR1340947
  4. G. G. Elder, Galois module structure of ambiguous ideals in biquadratic extensions. Can. J. Math. 50 (1998), 1007–1047. Zbl1015.11056MR1650942
  5. G. G. Elder, On the Galois structure of the integers in cyclic extensions of local number fields. J. Théor. Nombres Bordeaux. 14 (2002), 113–149. Zbl1026.11083MR1925994
  6. G. G. Elder, The Galois module structure of ambiguous ideals in cyclic extensions of degree 8 . To appear in the Proceedings of the International Algebraic Conference dedicated to the memory of Z. I. Borevich, Sept 17–23, 2002. 
  7. R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science. Addison Wesley, Reading MA 1989. Zbl0668.00003MR1397498
  8. J. V. Kuzmin, Representations of finite groups by automorphisms of nilpotent near spaces and by automorphisms of nilpotent groups. Sibirsk. Mat. Ž. 13 (1972), 107–117. Zbl0229.20007MR369505
  9. J-P. Serre, Local Fields. Springer-Verlag, New York, 1979. Zbl0423.12016MR554237
  10. A. Weiss, Rigidity of p -adic p -torsion. Ann. of Math. (2) 127 (1988), 317–332. Zbl0647.20007MR932300
  11. B. Wyman, Wildly ramified gamma extensions. Amer. J. Math. 91 (1969), 135–152. Zbl0188.11003MR241386

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