On fields of algebraic numbers with bounded local degrees

Checcoli, Sara; Zannier, Umberto

doi:10.1016/j.crma.2010.12.007

Cited by 7 publications

(26 citation statements)

References 11 publications

(18 reference statements)

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“…These results were established by the first author and U. Zannier in the situation F/K(B) is a number field extension F/K [CZ11], [Che11]. It turns out that the core of their arguments is the Tchebotarev property that we have identified.…”

Section: Introductionsupporting

confidence: 57%

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Tchebotarev theorems for function fields

Checcoli¹,

Dèbes

2016

Journal of Algebra

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We prove Tchebotarev type theorems for function field extensions over various base fields: number fields, finite fields, padic fields, PAC fields, etc. The Tchebotarev conclusion -existence of appropriate cyclic residue extensions -also compares to the Hilbert specialization property. It is more local but holds in more situations and extends to infinite extensions. For a function field extension satisfying the Tchebotarev conclusion, the exponent of the Galois group is bounded by the l.c.m. of the local specialization degrees. Further local-global questions arise for which we provide answers, examples and counter-examples.

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Section: Introductionsupporting

confidence: 57%

“…However the answer to the question is "Yes" if the group G is abelian. For number field extensions this was first proved in [CZ11].…”

Section: A Refined Question a Special Situation Where The Exponent Ofmentioning

confidence: 90%

Tchebotarev theorems for function fields

Checcoli¹,

Dèbes

2016

Journal of Algebra

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“…On the other hand it is an open question whether every field in Q with uniformly bounded local degrees has property (N). Recently Zannier and the first author (Theorem 1.1 in [4]) proved the existence of subfields of Q with uniformly bounded local degrees that are not contained in Q (d) for any d.…”

Section: Questionmentioning

confidence: 99%

“…Theorem 2 (Checcoli, 2011). An algebraic Galois extension of the rationals has uniformly bounded local degrees if and only if its Galois group has finite exponent.…”

Section: Introductionmentioning

confidence: 96%

On the Northcott property and other properties related to polynomial mappings

Checcoli

Widmer

2013

Math. Proc. Camb. Phil. Soc.

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We prove that if K/Q is a Galois extension of finite exponent andis the compositum of all extensions of K of degree at most d, then K (d) has the Bogomolov property and the maximal abelian subextension of K (d) /Q has the Northcott property.Moreover, we prove that given any sequence of finite solvable groups {Gm}m there exists a sequence of Galois extensions {Km}m with Gal(Km/Q) = Gm such that the compositum of the fields Km has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in Q (d) .We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewicz.

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“…In [CZ11] and [Che13] it is proved that a Galois extension of a number field has uniformly bounded local degrees if and only if its Galois group has finite exponent. So, for a Galois extension, the uniform boundedness of the local degrees translates into a property of its Galois group.…”

Section: Introductionmentioning

confidence: 99%

A note on Galois groups and local degrees

Checcoli

2018

manuscripta math.

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In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if K is a number field and L/K is an infinite Galois extension of group G, then the local degrees of L are uniformly bounded at all rational primes if and only if G has finite exponent. In this note we show that the non uniform boundedness of the local degrees is not equivalent to any group theoretical property. More precisely, we exhibit several groups that admit two different realisations over a given number field, one with bounded local degrees at a given set of primes and one with infinite local degrees at the same primes.

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On fields of algebraic numbers with bounded local degrees

Cited by 7 publications

References 11 publications

Tchebotarev theorems for function fields

Tchebotarev theorems for function fields

On the Northcott property and other properties related to polynomial mappings

A note on Galois groups and local degrees

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