Local certification of residual squareclasses in $\mathbb Q(\sqrt{2},\sqrt{pq},\sqrt{ps})$: one-bit, affine, and finite-choice Hilbert-symbol frameworks

Abstract

Recent works of El Hamam described explicit fundamental systems of units for several families of multiquadratic fields of degrees 8 and 16. In the degree-8 field $L^+ = \mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}),$ the corrected classification still leaves a residual binary indeterminacy: one must decide which of two explicitly constructed squareclasses gives the final unit generator. In this paper, we make this remaining bit explicit. First, we give an explicit local criterion deciding the parameter $μ\in \{1, ε_{pq}\}$ left open in recent literature. The criterion is first expressed in terms of Hilbert symbols at a single finite place, and is then sharpened to a residue criterion at a chosen split auxiliary rational prime. Second, we show that the standard residue datum $D(p,q,s) = \left( p \bmod 8,\,\, q \bmod 8,\,\, s \bmod 8,\,\, \biggl(\dfrac{q}{p}\biggr),\,\, \biggl(\dfrac{s}{p}\biggr),\,\, \biggl(\dfrac{q}{s}\biggr) \right)$ does not determine the final generator: we compute explicit triples with the same $D(p,q,s)$ but opposite values of the residual bit. Third, we place the one-bit problem inside a hierarchy of local-certification results in $K^\times/K^{\times2}$: besides the linear residual-choice statement, we prove an affine local-certification theorem for residual-choice cosets and a finite-test-set separation theorem for arbitrary finite candidate families.