Field Arithmetic

A computably presented algebraic field F has a splitting algorithm if it is decidable which polynomials in F [X] are irreducible there. We prove that such a field is computably categorical iff it is decidable which pairs of elements of F belong to the same orbit under automorphisms. We also show that this criterion is equivalent to the relative computable categoricity of F . 3955 Licensed to Univ of Mississippi. Prepared on Tue Jul 14 02:15:54 EDT 2015 for download from IP 130.74.92.202.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3956 RUSSELL MILLER AND ALEXANDRA SHLAPENTOKH versions essentially study how much information about the structures is needed to compute isomorphisms.Among these, computable categoricity remains the most widely studied concept. It is often equivalent to relative computable categoricity, but exceptions are known to exist; see [15] for an exception, and [9] for conditions implying equivalence. Traditionally, the main question has been to determine, for a particular class of structures, some structural criterion which is equivalent to computable categoricity. In early examples, from around 1980, Dzgoev, Goncharov, and Remmel showed (independently; see [10, 24]) that a linear order is computably categorical iff it has only finitely many pairs of adjacent elements, and Remmel also showed in [25] that a Boolean algebra is computably categorical iff it has only finitely many atoms. In both cases, the structural criterion identifies the obstacle to computing isomorphisms, in the class of structures under consideration. On the other hand, the criteria equivalent to computable categoricity for trees (viewed either as partial orders, or under the meet relation, and also with distinguished subtrees), established in [16] and [18] by Lempp, McCoy, Miller, and Solomon and in [13] by Kogabaev, Kudinov, and Miller, are not easy to describe in any known way, even though they are "structural", in any reasonable sense of the word. In terms of computational complexity, they are Σ 0 3 , as defined in Section 2, just like the conditions for linear orders and Boolean algebras. Indeed, for all of these structures, computable categoricity is equivalent to relative computable categoricity, and it is known from work by Ash, Knight, Manasse, and Slaman in [1], and independently by Chisholm in [2], that the computational complexity of relative computable categoricity is always Σ 0 3 . Our intention in this paper is to give a criterion for computable categoricity for algebraic fields with splitting algorithms. This should be viewed as a first step towards a criterion for computable fields in general, for which the question of computable categoricity has long been studied and has proven highly intractable. The basic definitions regarding computable fields appear in Section 3. Apart from algebraically closed fields, we believe that ours is the first result to characterize computable categoricity for any natural class of fields. (An algebraically closed fie...

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“…and so we can find y, with the uniformities described above. For a proof giving an actual formula for the generator y, see [6].…”

Section: Useful Results On Fieldsmentioning

confidence: 99%

Computable categoricity for algebraic fields with splitting algorithms

Miller

Shlapentokh

2014

Trans. Amer. Math. Soc.

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“…Then G(F 0 ) ≃Ẑ, and A contains F 0 . Furthermore, because G(F 0 ) is isomorphic toẐ, there is a pseudo-finite field F which is a regular extension of F 0 (this follows easily from Thm 23.1.1 in [3]), so that the restriction map G(F ) → G(F 0 ) is an isomorphism. By Corollary 3.1 in [4], the theory of F eliminates imaginaries in the language augmented by constants for elements of F 0 .…”

Section: Commentmentioning

confidence: 97%

Geometric Representation in the Theory of Pseudo-Finite Fields

Beyarslan¹,

Chatzidakis²

2017

J. symb. log.

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“…Suppose that S is thin. By [FJ05,p. 245 Claim B(f)] any set of type C 1 is contained in a set of type C 2 , so we have that…”

Section: Closed Points Of Arithmetic Descent For Cyclic Groupsmentioning

confidence: 98%

Arithmetic descent of specializations of Galois covers

Eberhart¹,

Hasson²

2017

Funct. Approx. Comment. Math.

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Abstract. Given a G-Galois branched cover of the projective line over a number field K, we study whether there exists a closed point of P 1 K with a connected fiber such that the G-Galois field extension induced by specialization "arithmetically descends" to Q (i.e., there exists a G-Galois field extension of Q whose compositum with the residue field of the point is equal to the specialization). We prove that the answer is frequently positive (whenever G is regularly realizable over Q) if one first allows a base change to a finite extension of K. If one does not allow base change, we prove that the answer is positive when G is cyclic. Furthermore, we provide an explicit example of a Galois branched cover of P 1 K with no K-rational points of arithmetic descent.

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Field Arithmetic

Cited by 104 publications

References 0 publications

Computable categoricity for algebraic fields with splitting algorithms

Computable categoricity for algebraic fields with splitting algorithms

Geometric Representation in the Theory of Pseudo-Finite Fields

Arithmetic descent of specializations of Galois covers

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