A subfield of the reals is constructed in which the set of squares, defined by a quantifier-free formula, is not Borel, and the field has the independence property.
Hieronymi , ‘Tameness beyond o-minimality’, Preprint, 2025, https://www.math.uni-bonn.de/people/ phierony/Tameness_Fields.pdf
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On Tameness, Measurability and the Independence Property
A subfield of the reals is constructed in which the set of squares, defined by a quantifier-free formula, is not Borel, and the field has the independence property.