On Galois structure of the integers in cyclic extensions of local number fields

G. Griffith Elder

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 1, page 113-149
  • ISSN: 1246-7405

Abstract

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Let p be a rational prime, K be a finite extension of the field of p -adic numbers, and let L / K be a totally ramified cyclic extension of degree p n . Restrict the first ramification number of L / K to about half of its possible values, b 1 > 1 / 2 · p e 0 / ( p - 1 ) where e 0 denotes the absolute ramification index of K . Under this loose condition, we explicitly determine the ℤ p [ G ] -module structure of the ring of integers of L , where ℤ p denotes the p -adic integers and G denotes the Galois group Gal ( L / K ) . In the process of determining this structure, we study various restrictions on the ramification filtration and examine the trace map relationships that result. Two of these restrictions are generalizations of almost maximal ramification. Our method for determining this structure is constructive (also inductive). We exhibit generators for the ring of integers of L over the group ring, ℤ p [ G ] (actually over 𝔒 T [ G ] where 𝔒 T is the ring of integers in the maximal unramified subfield of K ). They are determined in an essential way by their valuation. Then we describe their relations.

How to cite

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Elder, G. Griffith. "On Galois structure of the integers in cyclic extensions of local number fields." Journal de théorie des nombres de Bordeaux 14.1 (2002): 113-149. <http://eudml.org/doc/248895>.

@article{Elder2002,
abstract = {Let $p$ be a rational prime, $K$ be a finite extension of the field of $p$-adic numbers, and let $L/K$ be a totally ramified cyclic extension of degree $p^n$. Restrict the first ramification number of $L/K$ to about half of its possible values, $b_1 &gt; 1/2 \cdot pe_0/(p-1)$ where $e_0$ denotes the absolute ramification index of $K$. Under this loose condition, we explicitly determine the $\mathbb \{Z\}_p[G]$-module structure of the ring of integers of $L$, where $\mathbb \{Z\}_p$ denotes the $p$-adic integers and $G$ denotes the Galois group Gal$(L/K)$. In the process of determining this structure, we study various restrictions on the ramification filtration and examine the trace map relationships that result. Two of these restrictions are generalizations of almost maximal ramification. Our method for determining this structure is constructive (also inductive). We exhibit generators for the ring of integers of $L$ over the group ring, $\mathbb \{Z\}_p[G]$ (actually over $\mathfrak \{O\}_T[G]$ where $\mathfrak \{O\}_T$ is the ring of integers in the maximal unramified subfield of $K$). They are determined in an essential way by their valuation. Then we describe their relations.},
author = {Elder, G. Griffith},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {113-149},
publisher = {Université Bordeaux I},
title = {On Galois structure of the integers in cyclic extensions of local number fields},
url = {http://eudml.org/doc/248895},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Elder, G. Griffith
TI - On Galois structure of the integers in cyclic extensions of local number fields
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 1
SP - 113
EP - 149
AB - Let $p$ be a rational prime, $K$ be a finite extension of the field of $p$-adic numbers, and let $L/K$ be a totally ramified cyclic extension of degree $p^n$. Restrict the first ramification number of $L/K$ to about half of its possible values, $b_1 &gt; 1/2 \cdot pe_0/(p-1)$ where $e_0$ denotes the absolute ramification index of $K$. Under this loose condition, we explicitly determine the $\mathbb {Z}_p[G]$-module structure of the ring of integers of $L$, where $\mathbb {Z}_p$ denotes the $p$-adic integers and $G$ denotes the Galois group Gal$(L/K)$. In the process of determining this structure, we study various restrictions on the ramification filtration and examine the trace map relationships that result. Two of these restrictions are generalizations of almost maximal ramification. Our method for determining this structure is constructive (also inductive). We exhibit generators for the ring of integers of $L$ over the group ring, $\mathbb {Z}_p[G]$ (actually over $\mathfrak {O}_T[G]$ where $\mathfrak {O}_T$ is the ring of integers in the maximal unramified subfield of $K$). They are determined in an essential way by their valuation. Then we describe their relations.
LA - eng
UR - http://eudml.org/doc/248895
ER -

References

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  11. [11] M. Rzedowski-Calderón, G. Villa-Salvador, M.L. Madan, Galois module structure of rings of integers. Math. Z.204 (1990), 401-424. Zbl0682.12003MR1107472
  12. [12] J-P. Serre, Local fields. Springer-Verlag, Berlin/Heidelberg/New York, 1979. Zbl0423.12016MR554237
  13. [13] S. Ullom, Integral Normal Bases in Galois Extensions of Local Fields. Nagoya Math. J.39 (1970), 141-148. Zbl0199.08401MR263790
  14. [14] S.V. Vostokov, Ideals of an Abelian p-extension of a local field as Galois modules. Zap. Naun. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 57 (1976), 64-84. Zbl0355.12012MR453708
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  16. [16] H. Yokoi, On the ring of integers in an algebraic number field as a representation module of galois group. Nagoya Math. J.16 (1960), 83-90. Zbl0119.27703MR123563

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