Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Explicit integral Galois module structure of weakly ramified extensions of local fields


Author: Henri Johnston
Journal: Proc. Amer. Math. Soc. 143 (2015), 5059-5071
MSC (2010): Primary 11R33, 11S15
DOI: https://doi.org/10.1090/proc/12634
Published electronically: May 7, 2015
MathSciNet review: 3411126
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ L/K$ be a finite Galois extension of complete local fields with finite residue fields and let $ G=\operatorname {Gal}(L/K)$. Let $ G_{1}$ and $ G_{2}$ be the first and second ramification groups. Thus $ L/K$ is tamely ramified when $ G_{1}$ is trivial and we say that $ L/K$ is weakly ramified when $ G_{2}$ is trivial. Let $ \mathcal {O}_{L}$ be the valuation ring of $ L$ and let $ \mathfrak{P}_{L}$ be its maximal ideal. We show that if $ L/K$ is weakly ramified and $ n \equiv 1 \bmod \vert G_{1}\vert$, then $ \mathfrak{P}_{L}^{n}$ is free over the group ring $ \mathcal {O}_{K}[G]$, and we construct an explicit generating element. Under the additional assumption that $ L/K$ is wildly ramified, we then show that every free generator of $ \mathfrak{P}_{L}$ over $ \mathcal {O}_{K}[G]$ is also a free generator of $ \mathcal {O}_{L}$ over its associated order in the group algebra $ K[G]$. Along the way, we prove a `splitting lemma' for local fields, which may be of independent interest.


References [Enhancements On Off] (What's this?)

  • [BE14] N. P. Byott and G. G. Elder, Sufficient conditions for large Galois scaffolds, http://arxiv.org/abs/1308.2092v2, 2014.
  • [Byo99] Nigel P. Byott, Integral Galois module structure of some Lubin-Tate extensions, J. Number Theory 77 (1999), no. 2, 252-273. MR 1702149 (2000f:11156), https://doi.org/10.1006/jnth.1999.2385
  • [Cha96] Robin J. Chapman, A simple proof of Noether's theorem, Glasgow Math. J. 38 (1996), no. 1, 49-51. MR 1373957 (97a:11186), https://doi.org/10.1017/S0017089500031244
  • [CO81] Lindsay N. Childs and Morris Orzech, On modular group rings, normal bases, and fixed points, Amer. Math. Monthly 88 (1981), no. 2, 142-145. MR 606253 (82j:12025), https://doi.org/10.2307/2321137
  • [CR81] Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 632548 (82i:20001)
  • [Ere91] B. Erez, The Galois structure of the square root of the inverse different, Math. Z. 208 (1991), no. 2, 239-255. MR 1128708 (92g:11108), https://doi.org/10.1007/BF02571523
  • [FT93] A. Fröhlich and M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, Cambridge, 1993. MR 1215934 (94d:11078)
  • [Has02] Helmut Hasse, Number theory, Translated from the third (1969) German edition, Classics in Mathematics, Springer-Verlag, Berlin, 2002. Reprint of the 1980 English edition [Springer, Berlin; MR0562104 (81c:12001b)]; Edited and with a preface by Horst Günter Zimmer. MR 1885791
  • [Joh14] H. Johnston, Iterated semi-direct products, MathOverflow, 2014, URL:
    http://mathoverflow.net/q/156209 (version: 2014-01-30).
  • [Kaw86] Fuminori Kawamoto, On normal integral bases of local fields, J. Algebra 98 (1986), no. 1, 197-199. MR 825142 (87e:11137), https://doi.org/10.1016/0021-8693(86)90022-0
  • [Köc04] Bernhard Köck, Galois structure of Zariski cohomology for weakly ramified covers of curves, Amer. J. Math. 126 (2004), no. 5, 1085-1107. MR 2089083 (2005i:11163)
  • [KS04] Hans Kurzweil and Bernd Stellmacher, The theory of finite groups, Universitext, Springer-Verlag, New York, 2004. An introduction; Translated from the 1998 German original. MR 2014408 (2004h:20001)
  • [Let98] Günter Lettl, Relative Galois module structure of integers of local abelian fields, Acta Arith. 85 (1998), no. 3, 235-248. MR 1627831 (99d:11127)
  • [Noe32] E. Noether, Normalbasis bei Körpern ohne höhere Verzweigung., J. Reine Angew. Math. 167 (1932), 147-152.
  • [Sem88] I. A. Semaev, Construction of polynomials, irreducible over a finite field, with linearly independent roots, Mat. Sb. (N.S.) 135(177) (1988), no. 4, 520-532, 560 (Russian); English transl., Math. USSR-Sb. 63 (1989), no. 2, 507-519. MR 942137 (89i:11135)
  • [Ser79] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237 (82e:12016)
  • [Tho08] Lara Thomas, A valuation criterion for normal basis generators in equal positive characteristic, J. Algebra 320 (2008), no. 10, 3811-3820. MR 2457723 (2009i:12007), https://doi.org/10.1016/j.jalgebra.2008.05.024
  • [Ull69a] S. Ullom, Galois cohomology of ambiguous ideals, J. Number Theory 1 (1969), 11-15. MR 0237473 (38 #5755)
  • [Ull69b] S. Ullom, Normal bases in Galois extensions of number fields, Nagoya Math. J. 34 (1969), 153-167. MR 0240082 (39 #1436)
  • [Ull70] S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J. 39 (1970), 141-148. MR 0263790 (41 #8390)
  • [Vin05] Stéphane Vinatier, Galois module structure in weakly ramified 3-extensions, Acta Arith. 119 (2005), no. 2, 171-186. MR 2167720 (2006d:11135), https://doi.org/10.4064/aa119-2-3
  • [Vos81] S. V. Vostokov, Normal basis for an ideal in a local ring, J. Sov. Math. 17 (1981), 1755-1758.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11R33, 11S15

Retrieve articles in all journals with MSC (2010): 11R33, 11S15


Additional Information

Henri Johnston
Affiliation: Department of Mathematics, University of Exeter, Exeter, EX4 4QF, United Kingdom
Email: H.Johnston@exeter.ac.uk

DOI: https://doi.org/10.1090/proc/12634
Received by editor(s): August 20, 2014
Received by editor(s) in revised form: September 15, 2014
Published electronically: May 7, 2015
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society