Galois structure of ideals in wildly ramified abelian p -extensions of a p -adic field, and some applications

Nigel P. Byott

Journal de théorie des nombres de Bordeaux (1997)

  • Volume: 9, Issue: 1, page 201-219
  • ISSN: 1246-7405

Abstract

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Let K be a finite extension of ℚ p with ramification index e , and let L / K be a finite abelian p -extension with Galois group Γ and ramification index p n . We give a criterion in terms of the ramification numbers t i for a fractional ideal 𝔓 h of the valuation ring S of L not to be free over its associated order 𝔄 ( K Γ ; 𝔓 h ) . In particular, if t n - [ t n / p ] < p n - 1 e then the inverse different can be free over its associated order only when t i ≡ - 1 (mod p n ) for all i . We give three consequences of this. Firstly, if 𝔄 ( K Γ ; S ) is a Hopf order and S is 𝔄 ( K Γ ; S ) -Galois then t i ≡ - 1 (mod p n ) for all i . Secondly, if K = k r L = k m + r are Lubin-Tate division fields, with m > r and k ≠ ℚ p , then S is not free over ( 𝔄 ( K Γ ; S ) . Thirdly, these extensions k m + r / k r admit two Hopf Galois structures exhibiting different behaviour at integral level.

How to cite

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Byott, Nigel P.. "Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications." Journal de théorie des nombres de Bordeaux 9.1 (1997): 201-219. <http://eudml.org/doc/247997>.

@article{Byott1997,
abstract = {Let $K$ be a finite extension of $\mathbb \{Q\}_p$ with ramification index $e$, and let $L/K$ be a finite abelian $p$-extension with Galois group $\Gamma $ and ramification index $p^n$. We give a criterion in terms of the ramification numbers $t_i$ for a fractional ideal $\mathfrak \{P\}^h$ of the valuation ring $S$ of $L$ not to be free over its associated order $\mathfrak \{A\} (K \Gamma ; \mathfrak \{P\}^h)$. In particular, if $t_n - [t_n/p] &lt; p^\{n-1\}e$ then the inverse different can be free over its associated order only when $t_i \equiv -1$ (mod $p^n$) for all $i$. We give three consequences of this. Firstly, if $\mathfrak \{A\} (K \Gamma ; S)$ is a Hopf order and $S$ is $\mathfrak \{A\} (K \Gamma ; S)$-Galois then $t_i \equiv -1$ (mod $p^n$) for all $i$. Secondly, if $K = k_r \, L = k_\{m+r\}$ are Lubin-Tate division fields, with $m &gt; r$ and $k \ne \mathbb \{Q\}_p$, then $S$ is not free over ($\mathfrak \{A\} (K \Gamma ; S)$. Thirdly, these extensions $k_\{m+r\}/k_r$ admit two Hopf Galois structures exhibiting different behaviour at integral level.},
author = {Byott, Nigel P.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Galois module structure; associated order; Hopf order; Lubin-Tate formal group; associated orders; Galois modules; ramification; Hopf Galois extensions; -adic field; finite abelian -extension; Kummer theory},
language = {eng},
number = {1},
pages = {201-219},
publisher = {Université Bordeaux I},
title = {Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications},
url = {http://eudml.org/doc/247997},
volume = {9},
year = {1997},
}

TY - JOUR
AU - Byott, Nigel P.
TI - Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications
JO - Journal de théorie des nombres de Bordeaux
PY - 1997
PB - Université Bordeaux I
VL - 9
IS - 1
SP - 201
EP - 219
AB - Let $K$ be a finite extension of $\mathbb {Q}_p$ with ramification index $e$, and let $L/K$ be a finite abelian $p$-extension with Galois group $\Gamma $ and ramification index $p^n$. We give a criterion in terms of the ramification numbers $t_i$ for a fractional ideal $\mathfrak {P}^h$ of the valuation ring $S$ of $L$ not to be free over its associated order $\mathfrak {A} (K \Gamma ; \mathfrak {P}^h)$. In particular, if $t_n - [t_n/p] &lt; p^{n-1}e$ then the inverse different can be free over its associated order only when $t_i \equiv -1$ (mod $p^n$) for all $i$. We give three consequences of this. Firstly, if $\mathfrak {A} (K \Gamma ; S)$ is a Hopf order and $S$ is $\mathfrak {A} (K \Gamma ; S)$-Galois then $t_i \equiv -1$ (mod $p^n$) for all $i$. Secondly, if $K = k_r \, L = k_{m+r}$ are Lubin-Tate division fields, with $m &gt; r$ and $k \ne \mathbb {Q}_p$, then $S$ is not free over ($\mathfrak {A} (K \Gamma ; S)$. Thirdly, these extensions $k_{m+r}/k_r$ admit two Hopf Galois structures exhibiting different behaviour at integral level.
LA - eng
KW - Galois module structure; associated order; Hopf order; Lubin-Tate formal group; associated orders; Galois modules; ramification; Hopf Galois extensions; -adic field; finite abelian -extension; Kummer theory
UR - http://eudml.org/doc/247997
ER -

References

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  1. [Be] A.-M. Bergé, Arithmétique d'une extension galoisienne à groupe d'inertie cyclique, Ann. Inst. Fourier, Grenoble28 (1978), 17-44. Zbl0377.12009MR513880
  2. [B-F] F. Bertrandias and M.-J. Ferton, Sur l'anneau des entiers d'une extension cyclique de degré premier d'un corps local, C. R. Acad. Sc. Paris274 (1972), A1330-A1333. Zbl0235.12007MR296047
  3. [Bl-Bu] W. Bley and D. Burns, Über arithmetische assoziierte Ordnungen, J. Number Theory58 (1996), 361-387. Zbl0857.11063MR1393621
  4. [Bu1] D. Burns, Factorisability and wildly ramified Galois extensions, Ann. Inst. Fourier, Grenoble41 (1991), 393-430. Zbl0727.11048MR1137291
  5. [Bu2] D. Burns, On the equivariant structure of ideals in Galois extensions of fields, Preprint, King's CollegeLondon (1996). 
  6. [By1] N.P. Byott, Some self-dual rings of integers not free over their associated orders, Math. Proc. Camb. Phil. Soc.110 (1991), 5-10; Corrigendum116 (1994), 569. Zbl0737.11037MR1104596
  7. [By2] N. Byott, On Galois isomorphisms between ideals in extensions of local fields, Manuscripta Math.73 (1991), 289-311. Zbl0771.11047MR1132141
  8. [By3] N.P. Byott, Tame and Galois extensions with respect to Hopf orders, Math. Z.220 (1995), 495-522. Zbl0841.16021MR1363852
  9. [By4] N.P. Byott, Uniqueness of Hopf Galois structure for separable field extensions, Comm. Alg.24 (1996), 3217-3228; Corrigendum24 (1996), 3705. Zbl0878.12001MR1402555
  10. [By5] N.P. Byott, Associated orders of certain extensions arising from Lubin- Tate formal groups, to appear in J. de Théorie des Nombres de Bordeaux. Zbl0902.11052MR1617408
  11. [By-L] N.P. Byott and G. Lettl, Relative Galois module structure of integers in abelian fields, J. de Théorie des Nombres de Bordeaux8 (1996), 125-141. Zbl0859.11059MR1399950
  12. [C-L] S.-P. Chan and C.-H. Lim, The associated orders of rings of integers in Lubin-Tate division fields over the p-adic number field, Ill. J. Math.39 (1995), 30-38. Zbl0816.11061MR1299647
  13. [C] L.N. Childs, Taming wild extensions with Hopf algebras, Trans. Am. Math. Soc.304 (1987), 111-140. Zbl0632.12013MR906809
  14. [C-M] L.N. Childs and D.J. Moss, Hopf algebras and local Galois module theory, in Advances in Hopf Algebras, Lect. Notes Pure and Appl. Math. Series, Vol. 158 (J. Bergen and S. Montgomery, eds.), Dekker, 1994, pp. 1-14. Zbl0826.16035MR1289419
  15. [E] G.G. Elder, Galois module structure of ideals in wildly ramified cyclic extensions of degree p2, Ann. Inst. Fourier, Grenoble45 (1995), 625-647. Zbl0820.11070MR1340947
  16. [E-M] G.G. Elder and M.L. Madan, Galois module structure of the integers in wildly ramified cyclic extensions, J. Number Theory47 (1994), 138-174. Zbl0801.11046MR1275759
  17. [F] M.-J. Ferton, Sur les idéaux d'une extension cyclique de degré premier d'un corps local, C. R. Acad. Paris276 (1973), A1483-A1486. Zbl0268.12006MR332733
  18. [G] C. Greither, Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring, Math. Z.210 (1992), 37-67. Zbl0737.11038MR1161169
  19. [G-P] C. Greither and B. Pareigis, Hopf Galois theory for separable field extensions, J. Algebra106 (1987), 239-258. Zbl0615.12026MR878476
  20. [L] H.-W. Leopoldt, Uber die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. reine u. angew. Math.201 (1959), 119-149. Zbl0098.03403MR108479
  21. [RC-VS-M] M. Rzedowski Calderón, G.D. Villa Salvador and M.L. Madan, Galois module structure of rings of integers, Math. Z.204 (1990), 401-424. Zbl0682.12003MR1107472
  22. [S1] J.-P. Serre, Local Class Field Theory, in Algebraic Number Theory (J.W.S. Cassels and A. Fröhlich, eds.), Academic Press, 1967. MR220701
  23. [S2] J.-P. Serre, Local fields (Graduate Texts in Mathematics, Vol. 67), Springer, 1979. Zbl0423.12016MR554237
  24. [T1] M.J. Taylor, Formal groups and the Galois module structure of local rings of integers, J. reine angew. Math.358 (1985), 97-103. Zbl0582.12008MR797677
  25. [T2] M.J. Taylor, Hopf structure and the Kummer theory of formal groups, J. reine angew. Math.375/376 (1987), 1-11. Zbl0609.12015MR882287
  26. [U] S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J.39 (1970), 141-148. Zbl0199.08401MR263790
  27. [V1] S.V. Vostokov, Ideals of an abelian p-extension of an irregular local field as Galois modules, Zap. Nauchn. Sem. Lening. Otdel. Math. Inst. Steklov. (LOMI) 46 (1974), 14-35; English transl. in J. Soviet Math.9 (1978), 299-317. Zbl0396.12016MR371865
  28. [V2] S.V. Vostokov, Ideals of an abelian p-extension of a local field as Galois module, Zap. Nauchn. Sem. Lening. Otdel. Math. Inst. Steklov. (LOMI) 57 (1976), 64-84; English transl. in J. Soviet Math.11 (1979), 567-584. Zbl0403.12017MR453708
  29. [W] W.C. Waterhouse, Normal basis implies Galois for coconnected Hopf algebras, Preprint, Pennsylvania State University (1992). 

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