Relative Galois module structure of rings of integers of absolutely abelian number fields

Johnston, Henri

doi:10.1515/crelle.2008.050

Cited by 5 publications

(5 citation statements)

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“…Moreover, Hom Λ (X, Y ) is a (left) End Λ (Y )-lattice in Hom A (KX, KY ) via post-composition. The following result is a generalisation of [Joh08, Proof. Let f ∈ Hom Λ (X, Y ).…”

Section: 2mentioning

confidence: 86%

Computing isomorphisms between lattices

Hofmann

Johnston

2020

Math. Comp.

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Let K be a number field, let A be a finite dimensional semisimple Kalgebra and let Λ be an O K -order in A. It was shown in previous work that, under certain hypotheses on A, there exists an algorithm that for a given (left) Λ-lattice X either computes a free basis of X over Λ or shows that X is not free over Λ. In the present article, we generalise this by showing that, under weaker hypotheses on A, there exists an algorithm that for two given Λ-lattices X and Y either computes an isomorphism X → Y or determines that X and Y are not isomorphic. The algorithm is implemented in Magma for A = Q[G] and Λ = Z[G], where G is a finite group satisfying certain hypotheses. This is used to investigate the Galois module structure of rings of integers and ambiguous ideals of tamely ramified Galois extensions of Q with Galois group isomorphic to Q 8 × C 2 , the direct product of the quaternion group of order 8 and the cyclic group of order 2. The first named author was supported by Project II.2 of SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German Research Foundation (DFG). The second named author was supported by EPSRC First Grant EP/N005716/1 'Equivariant Conjectures in Arithmetic'. Preliminaries on lattices and ordersFor further background on lattices and orders, we refer the reader to [Rei03, §4 and §8]. Let R be an integral domain with field of fractions K. To avoid trivialities, we assume that R = K. An R-lattice is a finitely generated torsion free module over R. For any finite dimensional K-vector space V , an R-lattice in V is a finitely generated R-submodule M in V . We define a K-vector subspace of V byNow further suppose that R is a noetherian integral domain and let A be a finite dimensional K-algebra. An R-order in A is a subring Λ of A (so in particular has the same unity element as A) such that Λ is a full R-lattice in A. Note that Λ is both left and right noetherian, since Λ is finitely generated over R. A left Λ-lattice X is a left Λ-module that is also an R-lattice; in this case, KX may be viewed as a left A-module.Henceforth all modules (resp. lattices) shall be assumed to be left modules (resp. lattices) unless otherwise stated. Two Λ-lattices are said to be isomorphic if they are isomorphic as Λ-modules.Lemma 2.1. Let S be a noetherian integral domain such that R ⊆ S ⊆ K. Let Γ be an S-order in A. Let V be a finitely generated A-module. For any R-lattice M in V , the setProof. That M ⊆ ΓM is clear. Note that K is the field of fractions of both R and S. Write M = v 1 , . . . , v l R and Γ = w 1 , . . . , w m S . An easy calculation shows thatand hence ΓM is an S-lattice in V . Moreover, it is straightforward to see that ΓM is also a Γ-module and therefore is a Γ-lattice in V .Lemma 2.2. Assume the setup and notation of Lemma 2.1. Let Λ be an R-order in A and let f : X → Y be a homomorphism of Λ-lattices. Then the following hold. 4 3.1. Restricting isomorphisms over maximal orders. By [Rei03, (10.4)] there exists a (not necessarily unique) maximal O-order M such that Λ ⊆ M ⊆ A...

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“…Moreover, Hom Λ (X, Y ) is a (left) End Λ (Y )-lattice in Hom A (KX, KY ) via post-composition. The following result is a generalisation of [Joh08, Proof. Let f ∈ Hom Λ (X, Y ).…”

Section: 2mentioning

confidence: 86%

Computing isomorphisms between lattices

Hofmann

Johnston

2020

Math. Comp.

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“…At the same time, Martinet proved that every quaternion extension of degree 8 over Q that is wildly ramified is such that O L is free over its associated order [123], which is not always true when the extension is tamely ramified [124]. [112] is interesting also because it gathers several properties of associated orders that might be very useful, some of them are originally issued from [48] and [57]. For example, the next two propositions show how associated orders in composite fields and subfields can be determined under certain additional assumptions, which sometimes permits the reduction of the problem to simpler extensions:…”

Section: Classical Galois Module Theory For Number Fieldsmentioning

confidence: 99%

“…[48], Lem. 6 -[112], Cor.2.5). -Let L/K and M/K be Galois extensions of number fields with K ⊂ M ⊂ L and L/M at most tamely ramified.…”

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confidence: 99%

On the Galois module structure of extensions of local fields

Thomas¹

2010

Publications Mathématiques De Besançon

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“…(A simplified proof of this result can be found in [18].) More generally, we say that a number field K is Leopoldt if, for every finite abelian extension L/K, the ring of integers O L is free over A L/K (note that this differs from the definition of Leopoldt given in [15]). Since A L/K = O K [G] if and only if L/K is tame, Leopoldt's Theorem implies the celebrated Hilbert-Speiser Theorem: Every tame finite abelian extension L of Q has a normal integral basis, that is, O L is free as a Z[G]-module.…”

Section: Introductionmentioning

confidence: 99%

On the restricted Hilbert–Speiser and Leopoldt properties

Byott¹,

Carter²,

Greither³

et al. 2011

Illinois J. Math.

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Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if, for every tame G-Galois extension L/K, the ring of integers O L is free as an O K [G]-module. If O L is free over the associated order A L/K for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-Speiser of type G. We show that the converse does not hold in general, but that a modified version does hold for many number fields K (in particular, for K/Q Galois) when G = C p has prime order. We give examples with G = C p to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

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Relative Galois module structure of rings of integers of absolutely abelian number fields

Abstract: Abstract. Let L/K be an extension of number fields where L/Q is abelian. We define such an extension to be Leopoldt if the ring of integers O L of L is free over the associated order A L/K . Furthermore we define an abelian number field K to be Leopoldt if every finite extension L/K

Cited by 5 publications

References 15 publications

Computing isomorphisms between lattices

Computing isomorphisms between lattices

On the Galois module structure of extensions of local fields

On the restricted Hilbert–Speiser and Leopoldt properties

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