{"id":"palomar/PALOMAR-2026-09-02-000014","url":"https://github.com/stalex444/golden-family-trace-forms/tree/9528df064df58318213b541048e05ff4c0d9cb67","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-02-000014-v1.json","repo":"https://github.com/stalex444/golden-family-trace-forms","tags":["math.NT","msc:11E04","msc:11E12","msc:15A63","msc:11R04","msc:11R16","msc:11R29","quadratic-forms","algebraic-number-theory","lean-formalization"],"score":0,"title":"The trace form of a tensor product of algebras and the signature product rule, with the trace forms of Q[x]/(x^3 - x - 1), Q[x]/(x^4 - x - 1) and their compositum as instances","trust":"high","domain":"math.NT","source":"palomar","authors":["Stephanie Alexander"],"reviews":0,"subject":"math.NT","summary":"GENERAL THEOREMS FIRST. For finite free algebras A, B over a commutative ring R, the trace form of the tensor product A ⊗_R B is the tensor product of the trace forms, and over a linearly ordered field the Sylvester signature of a tensor product of quadratic forms obeys a product rule. Seven compared general statements: (1) the trace of a pure tensor is the product of the traces, Tr_{A⊗B}(a ⊗ b) = Tr_A(a)·Tr_B(b); (2) the trace form on pure tensors is the product of the trace forms; (3) the Gram matrix of a tensor basis bA ⊗ bB (Mathlib's Basis.tensorProduct) is the Kronecker product of the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-02","reactions":{},"formalized":true}