minpoly — Mathlib

English

Evaluating minpoly at x via the subfield map gives zero.

Русский

Оценка minpoly в x через отображение подполе равно нулю.

LaTeX

$$$ \\mathrm{Polynomial.eval₂} (\\mathrm{Subfield.subtype} (\\mathrm{FixedPoints.subfield G F})) x (\\mathrm{minpoly} G F x) = 0 $$$

Lean4

/-- `minpoly G F x` is the minimal polynomial of `(x : F)` over `FixedPoints.subfield G F`. -/
def minpoly : Polynomial (FixedPoints.subfield G F) :=
  (prodXSubSMul G F x).toSubring (FixedPoints.subfield G F).toSubring fun _ hc g =>
    let ⟨n, _, hn⟩ := Polynomial.mem_coeffs_iff.1 hc
    hn.symm ▸ prodXSubSMul.coeff G F x g n

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