Fields of algebraic numbers with bounded local degrees and their properties

Checcoli, Sara

doi:10.1090/s0002-9947-2012-05712-6

Cited by 15 publications

(23 citation statements)

References 10 publications

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“…Another important ingredient in the proof of Theorem 1 is that the finite subextensions of Q (d) ab /Q can be generated by elements of uniformly bounded degree which is not true for Q (d) , provided d is large enough, as shown recently by the first author (Theorem 2 in [5]). Again this is not a necessary condition for property (N) (again see Corollary 2 in [19]).…”

Section: Questionmentioning

confidence: 99%

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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Section: Questionmentioning

confidence: 99%

On the Northcott property and other properties related to polynomial mappings

Checcoli

Widmer

2013

Math. Proc. Camb. Phil. Soc.

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“…We start with the more straightforward examples. More examples of such fields can be found in [1]. Most of such examples where the field is Galois over Q were already covered by definability results of Videla with respect to the ring of integers.…”

Section: Examples Of Infinite Extensions Of Q Where the Ring Of Integmentioning

confidence: 99%

First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers

Shlapentokh

2018

Isr. J. Math.

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We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold. (1) For any rational prime q and any positive rational integer m, algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by q m . (2) Given a prime q, and an integer m > 0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set {ξ p ℓ |ℓ ∈ Z >0 , p = q is any prime such that q m+1 |(p − 1)}. (3) The first-order theory of any abelian extension of Q with finitely many ramified rational primes is undecidable. We also show that under a condition on the splitting of one rational prime in an infinite algebraic extension of Q, the existence of a finitely generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is indecidable. produced a uniform definition of Z over rings of integers of number fields, R. Rumely in [30] improved J. Robinson's result making the definition of the ring of integers over number fields uniform across fields. More recently, B. Poonen in [22] and J. Koenigsmann in [12] updated J. Robinson's definition of integers by reducing the number of universal quantifiers used in these definitions, B. Poonen to two and J. Koenigsmann to one.The desire to reduce the number of universal quantifiers is motivated to large extent by the interest in extending Hilbert's Tenth Problem to Q. This would be accomplished by a purely existential definition of Z over Q. Unfortunately there are serious doubts as to whether such a definition exists. See [6], [21] and [33] for surveys on Hilbert's Tenth Problem and related questions of definability.A lot of work aiming to prove the decidability of the first-order theory has centered around various infinite extensions of Q. (See [6] for a survey of these results.) One of the more influential results was arguably due to R. Rumely in [31], where he showed that Hilbert's Tenth Problem is decidable over the ring of all algebraic integers. This result was strengthened by L. van den Dries proving in [40] that the first-order theory of this ring was decidable. Another remarkable result is due to M. Fried, D. Haran and H. Völklein in [10], where it is shown that the first-order theory of the field of all totally real algebraic numbers is decidable. This field constitutes a boundary of sorts between the "decidable" and "undecidable", since J. Robinson showed in [27] that the first-order theory of the ring of all totally real integers is undecidable. In the same paper, she also proved that the first-order theory of a family of totally real rings of integers is undecidable and produced a "blueprint" for such proofs over ring...

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“…The following lemma can be found in [5] (cf. Proposition 2.4), and follows from some basic facts about the extraspecial p-groups (see for example [1] and [8]).…”

Section: Unboundedness: Proofs Of Theorems 4 Andmentioning

confidence: 99%

On the compositum of all degree d extensions of a number field

Gal¹,

Grizzard²

2014

J. Théor. Nombres Bordeaux

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Let k be a number field, and denote by k^[d] the compositum of all degree d extensions of k in a fixed algebraic closure. We first consider the question of whether all algebraic extensions of k of degree less than d lie in k^[d]. We show that this occurs if and only if d < 5. Secondly, we consider the question of whether there exists a constant c such that if K/k is a finite subextension of k^[d], then K is generated over k by elements of degree at most c. This was previously considered by Checcoli. We show that such a constant exists if and only if d < 3. This question becomes more interesting when one restricts attention to Galois extensions K/k. In this setting, we derive certain divisibility conditions on d under which such a constant does not exist. If d is prime, we prove that all finite Galois subextensions of k^[d] are generated over k by elements of degree at most d.Comment: 14 pages, 2 figure

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Fields of algebraic numbers with bounded local degrees and their properties

Cited by 15 publications

References 10 publications

On the Northcott property and other properties related to polynomial mappings

On the Northcott property and other properties related to polynomial mappings

First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers

On the compositum of all degree d extensions of a number field

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