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

Journal de théorie des nombres de Bordeaux (2002)

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

## Access Full Article

top## Abstract

top## How to cite

topElder, 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, $b_1 > 1/2 \cdot pe_0/(p-1)$ 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, $b_1 > 1/2 \cdot pe_0/(p-1)$ 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

top- [1] F. Bertrandias, Sur les extensions cycliques de degré pn d'un corps local. Acta Arith.34 (1979), no. 4, 361-377. Zbl0381.12008MR543208
- [2] C.W. Curtis, I. Reiner, Methods of Representation Theory, vol. 1. Wiley-Interscience, New York, 1990. Zbl0616.20001MR1038525
- [3] G.G. Elder, Galois module structure of integers in wildly ramified cyclic extensions of degree p2. Ann. Inst. Fourier45 (1995), 625-647, errata ibid.48 (1998), 609-610. Zbl0820.11070MR1340947
- [4] G.G. Elder, Galois module structure of ideals in wildly ramified biquadratic extensions. Can. J. Math.50 (1998), 1007-1047. Zbl1015.11056MR1650942
- [5] G.G. Elder, M.L. Madan, Galois module structure of integers in wildly ramified cyclic extensions. J. Number Theory47 (1994), 138-174. Zbl0801.11046MR1275759
- [6] A. Heller, I. Reiner, Representations of cyclic groups in rings of integers I. Ann. of Math. (2) 76 (1962), 73-92. Zbl0108.03101MR140575
- [7] E. Maus, Existenz p-adischer zahlkorper zu vorgegebenem verzweigungsverhalten. Ph.D. thesis, Univ. Hamburg, 1965.
- [8] H. Miki, On the ramification numbers of cyclic p-extensions over local fiels. J. Reine Angew. Math.328 (1981), 99-115. Zbl0457.12005MR636198
- [9] Y. Miyata, On the module structure in a cyclic extensions over a p-adic number field. Nagoya Math. J.73 (1979), 61-68. Zbl0402.12013MR524008
- [10] E. Noether, Normalbasis bei Körpern ohne höhere Verzweigung. J. Reine Angew. Math.167 (1932), 147-152. Zbl0003.14601JFM58.0172.02
- [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] J-P. Serre, Local fields. Springer-Verlag, Berlin/Heidelberg/New York, 1979. Zbl0423.12016MR554237
- [13] S. Ullom, Integral Normal Bases in Galois Extensions of Local Fields. Nagoya Math. J.39 (1970), 141-148. Zbl0199.08401MR263790
- [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
- [15] B. Wyman, Wildly ramified gamma extensions. Amer. J. Math.91 (1969), 135-152. Zbl0188.11003MR241386
- [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