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
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…
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