coeff_zero_minpoly — Mathlib

English

If R is maximal, there exists a valuation ring whose local subring equals R.

Русский

Если R является максимальным, существует valuationRing B со связью B.toLocalSubring = R.

LaTeX

$$$\\exists B:\\mathrm{ValuationSubring}(K), \\; B^{\\mathrm{toLocalSubring}} = R.$$$

Lean4

/-- For `x ∈ K` the valuation of the zeroth coefficient of the minimal polynomial
of `algebra_map K L x` over `K` is equal to the valuation of `x`. -/
@[simp]
theorem coeff_zero_minpoly (x : K) : v ((minpoly K (algebraMap K L x)).coeff 0) = v x := by
  rw [minpoly.eq_X_sub_C, coeff_sub, coeff_X_zero, coeff_C_zero, zero_sub, Valuation.map_neg]

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