Hermite's signature theorem and the coprime compositum, with the degree-12 compositum of x^3 - x - 1 and x^4 - x - 1
Kernel-checked proofs, in Lean 4 + Mathlib, of six general theorems and twelve statements about one number field. General: (1) Hermite's signature theorem for every real polynomial f, degenerate case included and with no monic hypothesis: the power-sum Hankel (Hermite) form H_f has sigPos = #R + #C+, sigNeg = #C+, rank sigPos + sigNeg = #D and radical of dimension deg f - #D, where D, R, C+ are the distinct complex roots, the real ones, and those in the upper half plane; (2) its number-field form (classical, Hermite-Sylvester; in number-field form credited to O. Taussky 1968): the trace form…
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