What this theme is
This theme gathers approaches that encode the zeros in algebraic objects: characteristic polynomials and determinants whose roots are the zeros, field towers and Galois extensions, tensor and matrix identities, and cohomological invariants borrowed from the function-field analogue of RH.
Why it recurs
The function-field Riemann Hypothesis was proved (Weil, Deligne) using algebraic geometry and cohomology. That success is a standing invitation to algebraize the number-field case, so determinant and field-tower constructions recur as attempts to import the winning strategy.
Relevance to the Riemann Hypothesis
If the zeros can be realized as roots of a determinant of a positive or unitary operator — or as eigenvalues forced real by an algebraic symmetry — the critical-line property becomes an algebraic identity. This mirrors how the function-field case was settled.