On Galois structure of the integers in cyclic extensions of local number fields
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 1, pp. 113-149.

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, ${b}_{1}>1/2·p{e}_{0}/\left(p-1\right)$ where ${e}_{0}$ denotes the absolute ramification index of $K$. Under this loose condition, we explicitly determine the ${ℤ}_{p}\left[G\right]$-module structure of the ring of integers of $L$, where ${ℤ}_{p}$ denotes the $p$-adic integers and $G$ denotes the Galois group Gal$\left(L/K\right)$. 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, ${ℤ}_{p}\left[G\right]$ (actually over ${𝔒}_{T}\left[G\right]$ 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.

Soit $p$ un nombre premier, $K$ une extension finie du corps des nombres $p$-adiques, et $L/K$ une extension cyclique ramifiée de degré ${p}^{n}$. On suppose que le premier nombre de ramification de $L/K$ satisfait ${b}_{1}>1/2·p{e}_{0}/\left(p-1\right)$${e}_{0}$ est l’indice de ramification absolu de $K$. Nous déterminons explicitement la structure de l’anneau des entiers de $L$ comme ${ℤ}_{p}\left[G\right]$-module, où ${ℤ}_{p}$ désigne l’anneau des entiers $p$-adiques et $G=\text{Gal}\left(L/K\right)$ le groupe de Galois de $L$.

@article{JTNB_2002__14_1_113_0,
author = {G. Griffith Elder},
title = {On {Galois} structure of the integers in cyclic extensions of local number fields},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {113--149},
publisher = {Universit\'e Bordeaux I},
volume = {14},
number = {1},
year = {2002},
zbl = {1026.11083},
mrnumber = {1925994},
language = {en},
url = {https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_113_0/}
}
TY  - JOUR
AU  - G. Griffith Elder
TI  - On Galois structure of the integers in cyclic extensions of local number fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2002
SP  - 113
EP  - 149
VL  - 14
IS  - 1
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_113_0/
UR  - https://zbmath.org/?q=an%3A1026.11083
UR  - https://www.ams.org/mathscinet-getitem?mr=1925994
LA  - en
ID  - JTNB_2002__14_1_113_0
ER  - 
%0 Journal Article
%A G. Griffith Elder
%T On Galois structure of the integers in cyclic extensions of local number fields
%J Journal de théorie des nombres de Bordeaux
%D 2002
%P 113-149
%V 14
%N 1
%I Université Bordeaux I
%G en
%F JTNB_2002__14_1_113_0
G. Griffith Elder. On Galois structure of the integers in cyclic extensions of local number fields. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 1, pp. 113-149. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_113_0/

[1] F. Bertrandias, Sur les extensions cycliques de degré pn d'un corps local. Acta Arith. 34 (1979), no. 4, 361-377. | MR | Zbl

[2] C.W. Curtis, I. Reiner, Methods of Representation Theory, vol. 1. Wiley-Interscience, New York, 1990. | MR | Zbl

[3] G.G. Elder, Galois module structure of integers in wildly ramified cyclic extensions of degree p2. Ann. Inst. Fourier 45 (1995), 625-647, errata ibid. 48 (1998), 609-610. | Numdam | MR | Zbl

[4] G.G. Elder, Galois module structure of ideals in wildly ramified biquadratic extensions. Can. J. Math. 50 (1998), 1007-1047. | MR | Zbl

[5] G.G. Elder, M.L. Madan, Galois module structure of integers in wildly ramified cyclic extensions. J. Number Theory 47 (1994), 138-174. | MR | Zbl

[6] A. Heller, I. Reiner, Representations of cyclic groups in rings of integers I. Ann. of Math. (2) 76 (1962), 73-92. | MR | Zbl

[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. | MR | Zbl

[9] Y. Miyata, On the module structure in a cyclic extensions over a p-adic number field. Nagoya Math. J. 73 (1979), 61-68. | MR | Zbl

[10] E. Noether, Normalbasis bei Körpern ohne höhere Verzweigung. J. Reine Angew. Math. 167 (1932), 147-152. | JFM | Zbl

[11] M. Rzedowski-Calderón, G. Villa-Salvador, M.L. Madan, Galois module structure of rings of integers. Math. Z. 204 (1990), 401-424. | MR | Zbl

[12] J-P. Serre, Local fields. Springer-Verlag, Berlin/Heidelberg/New York, 1979. | MR | Zbl

[13] S. Ullom, Integral Normal Bases in Galois Extensions of Local Fields. Nagoya Math. J. 39 (1970), 141-148. | MR | Zbl

[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. | MR | Zbl

[15] B. Wyman, Wildly ramified gamma extensions. Amer. J. Math. 91 (1969), 135-152. | MR | Zbl

[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. | MR | Zbl