# 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

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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

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