valuation_X_eq_neg_one — Mathlib

English

Given f ∈ Polynomial K and g ∈ Polynomial K with g ≠ 0, the valuation of RatFunc.mk f g equals intValuation f divided by intValuation g.

Русский

При f ∈ Polynomial K и g ∈ Polynomial K, g ≠ 0, valuation RatFunc.mk f g равна intValuation f / intValuation g.

LaTeX

$$$ \operatorname{valuation}(\mathrm{RatFunc K}) (\mathrm{RatFunc.mk } f g) = \frac{\mathrm{intValuation} f}{\mathrm{intValuation} g} $$$

Lean4

@[simp]
theorem valuation_X_eq_neg_one : (idealX K).valuation (RatFunc K) RatFunc.X = WithZero.exp (-1 : ℤ) :=
  by
  rw [← RatFunc.algebraMap_X, valuation_of_algebraMap, intValuation_singleton, WithZero.exp]
  · exact Polynomial.X_ne_zero
  · exact idealX_span K

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