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].

scholarly journals Galois module structure of units in real biquadratic number fields

2004 ◽  
Vol 111 (2) ◽  
pp. 105-124 ◽  
Author(s):  
Marcin Mazur ◽  
Stephen V. Ullom
Keyword(s):  
Number Fields ◽  
Module Structure ◽  
Galois Module ◽  
Galois Module Structure

Galois Module Structure of Ambiguous Ideals in Biquadratic Extensions

1998 ◽  
Vol 50 (5) ◽  
pp. 1007-1047
Author(s):  
G. Griffith Elder
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Indecomposable Module ◽  
Module Structure ◽  
Galois Module ◽  
Algebraic Number Fields ◽  
Galois Module Structure ◽  
Weak Restriction

AbstractLet N/K be a biquadratic extension of algebraic number fields, and G = Gal(N/K). Under a weak restriction on the ramification filtration associated with each prime of K above 2, we explicitly describe the ℤ[G]-module structure of each ambiguous ideal of N. We find under this restriction that in the representation of each ambiguous ideal as a ℤ[G]-module, the exponent (or multiplicity) of each indecomposable module is determined by the invariants of ramification, alone.For a given group, G, define SG to be the set of indecomposable ℤ[G]-modules, M, such that there is an extension, N/K, for which G ≅ Gal(N/K), and M is a ℤ[G]-module summand of an ambiguous ideal of N. Can SG ever be infinite? In this paper we answer this question of Chinburg in the affirmative.

Integral Representation of P-Class Groups In ℤp-Extensions and the Jacobian Variety

1998 ◽  
Vol 50 (6) ◽  
pp. 1253-1272 ◽  
Author(s):  
López-Bautista Pedro Ricardo ◽  
Gabriel Daniel Villa-Salvador
Keyword(s):  
Galois Group ◽  
Algebraic Function ◽  
Number Fields ◽  
Module Structure ◽  
Closed Field ◽  
Galois Module ◽  
Jacobian Variety ◽  
Galois Module Structure ◽  
Iwasawa Invariants ◽  
Cyclotomic Number Fields

AbstractFor an arbitrary finite Galois p-extension L/K of ℤp-cyclotomic number fields of CM-type with Galois group G = Gal(L/K) such that the Iwasawa invariants are zero, we obtain unconditionally and explicitly the Galois module structure of CL-(p), the minus part of the p-subgroup of the class group of L. For an arbitrary finite Galois p-extension L/K of algebraic function fields of one variable over an algebraically closed field k of characteristic p as its exact field of constants with Galois group G = Gal(L/K) we obtain unconditionally and explicitly the Galois module structure of the p-torsion part of the Jacobian variety JL(p) associated to L/k.

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