Paper051_Linear_Algebra_Done_Integer

We prove that the entire content of linear algebra—and therefore, by the Master Ledger (Paper 50), the entire content of mathematics—lives over the integers ℤ and their field of fractions ℚ. Every other field used in mathematics (ℝ, ℂ, 𝔽p, ℚp, Q̄, function fields) is either a quotient of ℤ, a completion of ℚ, or a convenience. No mathematical content is lost by working over ℤ. The real numbers add nothing that ℚ does not already contain as rational approximants with explicit error bounds; the complex numbers add nothing that ℚ[i] or algebraic closures of ℚ do not already provide as finite extensions; the finite fields 𝔽p = ℤ/pℤ are quotients of ℤ by definition. The Smith Normal Form over ℤ replaces diagonalization over fields. The characteristic polynomial has integer coefficients. The minimal polynomial has integer coefficients. The eigenvalues are algebraic numbers—roots of integer polynomials. There is no mathematical object that requires a transcendental number, an uncountable field, or a non-constructive existence proof. Linear algebra done right is linear algebra done integer.