Squares in arithmetic progression over certain non-primitive quartic number fields

TL;DR

This paper characterizes non-constant arithmetic progressions of squares over quadratic extensions of quadratic fields, providing conditions on the square-free integer D for such progressions to exist.

Contribution

It offers a new characterization of arithmetic progressions of squares over certain non-primitive quartic number fields, extending previous results in quadratic fields.

Findings

Identifies specific conditions on D for progressions to exist.

Provides a classification of such progressions over quadratic extensions.

Extends known results from quadratic to certain quartic fields.

Abstract

Let $D$ be a square-free integer. Under certain conditions on $D$, we characterize non-constant arithmetic progressions of squares over quadratic extensions of $\mathbb{Q}(\sqrt{D})$.