ne_zero_of_injective — Mathlib

English

For an integral element in an integrally closed domain, the minimal polynomial is irreducible and prime.

Русский

Для интегрального элемента в интегрально замкнутом домене минимальный полином иррационален и прост.

LaTeX

$$$\text{Prime}(minpoly\ R\ x)$$$

Lean4

/-- A variant of `IsConjRoot.ne_zero`, only assuming `Nontrivial R`,
`NoZeroSMulDivisors R A` and `Function.Injective (algebraMap R A)` instead of `Field R`. If `y` is
a conjugate root of a nonzero element `x`, then `y` is not zero.
-/
theorem ne_zero_of_injective [Nontrivial R] [NoZeroSMulDivisors R S] {x y : S} (hx : x ≠ 0) (h : IsConjRoot R x y)
    (hf : Function.Injective (algebraMap R S)) : y ≠ 0 := fun g => hx (eq_zero_of_injective (g ▸ h.symm) hf)

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