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Level 2 — ℚ̄. Exact algebraic numbers as (minimal polynomial, isolating interval) pairs.
We never store the decimal expansion of √2: we store that it is the unique
root of x² - 2 in (1, 2). Arithmetic on algebraic numbers is done with
resultants — the sum α+β is a root of Res_y(p(y), q(x-y)) and the
product is a root of Res_y(p(y), yᵈ q(x/y)) — followed by factorization
over ℚ and re-isolation of the relevant root.
Resultants are computed as Sylvester-matrix determinants over the polynomial ring ℚ[x], using the fraction-free Bareiss algorithm so every intermediate division is exact.
Structs§
- Algebraic
Number - An exact real algebraic number (Level 2).
- Polynomial
- A univariate polynomial with exact rational coefficients.