## Abstract

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

Moreover, let $\tilde{\mathbb{Q}}$ be the field of all algebraic numbers and let $\text{Gal}(\mathbb{Q})=\text{Gal}(\tilde{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. For each positive integer $e$ we equip the group $\text{Gal}{(\mathbb{Q})}^{e}$ with its unique normalized Haar measure. For each $\mathit{\sigma}=({\sigma}_{1},\dots ,{\sigma}_{e})\in \text{Gal}{(\mathbb{Q})}^{e}$ let $\tilde{\mathbb{Q}}(\mathit{\sigma})$ be the fixed field of ${\sigma}_{1},\dots ,{\sigma}_{e}$ in $\tilde{\mathbb{Q}}$ and let $\tilde{\mathbb{Z}}(\mathit{\sigma})$ be the ring of integers of $\tilde{\mathbb{Q}}(\mathit{\sigma})$. Given a sentence $\theta $ in the language of rings, we let $\alpha $ be the Haar measure of the set of all $\mathit{\sigma}\in \text{Gal}{(\mathbb{Q})}^{e}$ for which $\theta $ holds in $\tilde{\mathbb{Z}}(\mathit{\sigma})$. We prove that $\alpha $ 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

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

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