The Arithmetic of Time: reversible units of a degree-nine field. The units of K = LM (L, M cubic) with rank-one Hermitian symbol are exactly those of trivial relative norm to L and to M, a group of rank two, and exactly those whose Schur multipliers run both ways
Kernel-checked proofs, in Lean 4 + Mathlib, of ten statements. Let K be a number field of degree 9 containing a totally real cubic field L and a cubic field M with one real place. Then K = LM has signature (3, 3), and its nine embeddings are the pairs (lambda, mu). A slot assignment, fixed by a labelling of L's embeddings and a non-real embedding mu_1 of M, places the nine conjugates of x in K in a 3 x 3 Hermitian array, the symbol of x; the symbol of x^-1 is its entrywise (Hadamard) inverse. (1) For every finite index type, a positive semidefinite complex matrix with no zero entry has…
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