minpoly_neg_splits — Mathlib

English

In a tower R ⊆ K ⊆ L with IsIntegral R x, if minpoly R x splits over L, then minpoly K x splits over L.

Русский

В тюрьме R ⊆ K ⊆ L, при IsIntegral R x, если minpoly R x распадается в L, то minpoly K x распадается в L.

LaTeX

$$$\\text{Splits}(\\operatorname{algebraMap} R L, \\minpoly R x) \\Rightarrow \\text{Splits}(\\operatorname{algebraMap} K L, \\minpoly K x)$$$

Lean4

theorem minpoly_neg_splits [Algebra K L] {x : L} (g : (minpoly K x).Splits (algebraMap K L)) :
    (minpoly K (-x)).Splits (algebraMap K L) := by
  rw [minpoly.neg]
  apply splits_mul _ _ g.comp_neg_X
  simpa only [map_pow, map_neg, map_one] using splits_C (algebraMap K L) ((-1) ^ (minpoly K x).natDegree)

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