Computing Associated Orders and Galois Generating Elements of Unit Lattices

Bley, Werner

doi:10.1006/jnth.1997.2050

Cited by 8 publications

(8 citation statements)

References 14 publications

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“…The combination of Theorem 2.8 and Lemma 2.9 of [Ble97] is essentially equivalent to Corollary 2.4 given here specialized to the abelian case.…”

Section: A Necessary and Sufficient Condition For Freenessmentioning

confidence: 90%

“…The method of this section together with Algorithm 3.1 can also be used to investigate the Galois module structure of units as in [Ble97]. For a number field L, write U L for the units of O L and μ(L) for the subgroup of roots of unity.…”

Section: Remark 43mentioning

confidence: 99%

“…We describe an algorithm which combines and contains all of the methods of [Ble97], [Bur00,Appendix] and [BE05, Lemma 3.1].…”

Section: Computing Associated Ordersmentioning

confidence: 99%

“…Other Galois modules to which the algorithm can be applied include the G-stable ideals of O L and, in certain cases, the torsion-free part of O × L . The algorithm can be thought of as a non-abelian, higher rank generalization of the one given in [Ble97]; though stated for the Galois module structure of units, this can be adapted to general modules for G abelian and d = 1 with relatively few changes. It is also worth noting that under the same restrictions on G and d, the algorithm in [BE05] computes the Picard group Pic(A) and solves the corresponding refined discrete logarithm problem, thus computing a generator if it exists.…”

Section: Introductionmentioning

confidence: 99%

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Computing generators of free modules over orders in group algebras

Bley

Johnston²

2008

Journal of Algebra

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Let E be a number field and G be a finite group. Let A be any O E -order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] ∼ = χ M χ is explicitly computable and each M χ is in fact a matrix ring over a field, this leads to an algorithm that either gives elements α 1 , . . . , α d ∈ X such that X = Aα 1 ⊕ · · · ⊕ Aα d or determines that no such elements exist.Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O L , the ring of algebraic integers of L, and A to be the associated order A(E [G]; O L ) ⊆ E [G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = Q.

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“…The combination of Theorem 2.8 and Lemma 2.9 of [Ble97] is essentially equivalent to Corollary 2.4 given here specialized to the abelian case.…”

Section: A Necessary and Sufficient Condition For Freenessmentioning

confidence: 90%

Section: Remark 43mentioning

confidence: 99%

“…We describe an algorithm which combines and contains all of the methods of [Ble97], [Bur00,Appendix] and [BE05, Lemma 3.1].…”

Section: Computing Associated Ordersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computing generators of free modules over orders in group algebras

Bley

Johnston²

2008

Journal of Algebra

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“…Moreover, there is a computational criterion due to Fröhlich (see also [3,Lemma 2.7]), which allows one to decide whether O L is locally free over A L/K (O L ). If this is the case, one can use the algorithms of this paper to compute the class of…”

Section: Introductionmentioning

confidence: 99%

Picard Groups and Refined Discrete Logarithms

Bley¹,

Endres²

2005

LMS J. Comput. Math.

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Let K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by O K , and A denotes any O K -order in K [G]. The paper describes an algorithm that explicitly computes the Picard group Pic(A), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of O L in Pic(O K [G]) can be numerically determined.

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On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)

Burns

2000

Comment. Math. Helv.

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Abstract. Let p be an odd rational prime and K a finite extension of Qp. We give a complete classification of those finite abelian extensions L/K in which any ideal of the valuation ring of L is free over its associated order in Qp[Gal(L/K)]. In an appendix W. Bley describes an algorithm which can be used to determine the structure of Galois stable ideals in abelian extensions of number fields. The algorithm is applied to give several new and interesting examples.Mathematics Subject Classification (1991). 11R33.

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Computing Associated Orders and Galois Generating Elements of Unit Lattices

Cited by 8 publications

References 14 publications

Computing generators of free modules over orders in group algebras

Computing generators of free modules over orders in group algebras

Picard Groups and Refined Discrete Logarithms

On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)

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