Undecidability of $\mathbb Q^{(2)}$

TL;DR

This paper proves that the field obtained by combining all quadratic extensions of the rational numbers has an undecidable first-order theory, highlighting fundamental limits in understanding its logical structure.

Contribution

It establishes the undecidability of the compositum of all quadratic extensions of nd, a significant result in number theory and logic.

Findings

The theory of nd^{(2)} is undecidable.

The result impacts the understanding of the logical complexity of algebraic number fields.

It connects field extensions with logical undecidability in model theory.

Abstract

It is shown that the compositum $ \mathbb Q^{(2)}$ of all degree 2 extensions of $\mathbb Q$ has undecidable theory.