## Journal of Symbolic Logic

### Decidable Fragments of Field Theories

Shih-Ping Tung

#### Abstract

We say $\varphi$ is an $\forall\exists$ sentence if and only if $\varphi$ is logically equivalent to a sentence of the form $\forall x\exists y \psi(x,y)$, where $\psi(x,y)$ is a quantifier-free formula containing no variables except $x$ and $y$. In this paper we show that there are algorithms to decide whether or not a given $\forall\exists$ sentence is true in (1) an algebraic number field $K$, (2) a purely transcendental extension of an algebraic number field $K$, (3) every field with characteristic 0, (4) every algebraic number field, (5) every cyclic (abelian, radical) extension field over $\mathbf{Q}$, and (6) every field.

#### Article information

Source
J. Symbolic Logic, Volume 55, Issue 3 (1990), 1007-1018.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183743401

Mathematical Reviews number (MathSciNet)
MR1071310

Zentralblatt MATH identifier
0724.03010

JSTOR
links.jstor.org

#### Citation

Tung, Shih-Ping. Decidable Fragments of Field Theories. J. Symbolic Logic 55 (1990), no. 3, 1007--1018. https://projecteuclid.org/euclid.jsl/1183743401