scholarly journals On Galois structure of the integers in elementary abelian extensions of local number fields

2007 ◽  
Vol 125 (2) ◽  
pp. 442-458
Author(s):  
Yoshimasa Miyata
Keyword(s):  
Number Fields ◽  
Abelian Extensions ◽  
Galois Structure ◽  
Local Number

Chinburg's Third Invariant in the Factorisability Defect Class Group

1994 ◽  
Vol 46 (2) ◽  
pp. 324-342
Author(s):  
D. Holland
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Real Field ◽  
Imaginary Quadratic Fields ◽  
Euler Systems ◽  
Abelian Extensions ◽  
Third Invariant ◽  
Galois Structure ◽  
Ideal Class Groups ◽  
Stark's Conjecture

AbstractChinburg's third invariant Ω(N/K, 3) ∊ C1(Z[Γ]) of a Galois extension N/K of number fields with group Γ is closely related to the Galois structure of unit groups and ideal class groups, and deep unsolved problems such as Stark's conjecture.We give a formula for Ω(N/K, 3) modulo D(ZΓ) in the factorisability defect class group, reminiscent of analytic class number formulas. Specialising to the case of an absolutely abelian, real field N, we give a natural conjecture in terms of Hecke factorisations which implies the vanishing of the invariant in the defect class group.We prove this conjecture when N has prime-power conductor using Euler systems of cyclotomic units, Ramachandra units and Hecke factorisation. This supports a general conjecture of Chinburg, which in our situation specialises to the statement that Ω(N/K, 3) = 0 for such extensions.We also develop a slightly extended version of Euler systems of units for general abelian extensions, which will be applied to abelian extensions of imaginary quadratic fields elsewhere

Dixon–Selberg summation over local number fields

1992 ◽  
Vol 33 (6) ◽  
pp. 2111-2114
Author(s):  
E. Thiran ◽  
J. Weyers
Keyword(s):  
Number Fields ◽  
Local Number

Biquadratic Extensions with One Break

2002 ◽  
Vol 45 (2) ◽  
pp. 168-179 ◽  
Author(s):  
Nigel P. Byott ◽  
G. Griffith Elder
Keyword(s):  
Number Fields ◽  
Module Structure ◽  
Galois Module ◽  
Galois Module Structure ◽  
Local Number

AbstractWe explicitly describe, in terms of indecomposable ℤ2[G]-modules, the Galois module structure of ideals in totally ramified biquadratic extensions of local number fields with only one break in their ramification filtration. This paper completeswork begun in [Elder: Canad. J.Math. (5) 50(1998), 1007–1047].

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