2019 PRIMITIVE RECURSIVE DECIDABILITY FOR LARGE RINGS OF ALGEBRAIC INTEGERS
Aharon Razon
Author Affiliations +
Albanian J. Math. 13(1): 3-93 (2019). DOI: 10.51286/albjm/1556711193

Abstract

Lou v. d. Dries proves in [Dri88] that the elementary theory Th(˜) of the ring ˜ of all algebraic integers is decidable. For a prime number p, let 𝔽p(t)˜ be the algebraic closure of 𝔽p(t) and denote the integral closure of 𝔽p[t] in 𝔽p(t)˜ by 𝔽p[t]˜. Lou v. d. Dries and Angus Macintyre prove in [DrM90] that Th(𝔽p[t]˜) is decidable. One of the main results of this work states that both Th(˜) and Th(𝔽p[t]˜) are primitive recursive.

Moreover, let ˜ be the field of all algebraic numbers and let Gal()=Gal(˜/) be the absolute Galois group of . For each positive integer e we equip the group Gal()e with its unique normalized Haar measure. For each σ=(σ1,,σe)Gal()e let ˜(σ) be the fixed field of σ1,,σe in ˜ and let ˜(σ) be the ring of integers of ˜(σ). Given a sentence θ in the language of rings, we let α be the Haar measure of the set of all σGal()e for which θ holds in ˜(σ). We prove that α is a rational number which can be effectively computed in a primitive recursive way. We prove a similar result also in the function field case.

Funding Statement

This work originates from the PhD thesis of the author from Tel Aviv University and carried out under the supervision of Prof. Moshe Jarden.

Citation

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Aharon Razon. "PRIMITIVE RECURSIVE DECIDABILITY FOR LARGE RINGS OF ALGEBRAIC INTEGERS." Albanian J. Math. 13 (1) 3 - 93, 2019. https://doi.org/10.51286/albjm/1556711193

Information

Published: 2019
First available in Project Euclid: 11 July 2023

Digital Object Identifier: 10.51286/albjm/1556711193

Subjects:
Primary: 12E30

Keywords: Galois stratification , PAC field over a subring , Primitive recursive decidability

Rights: Copyright © 2019 Research Institute of Science and Technology (RISAT)

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