Theorems · Theorem · commutative algebra
MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent
∀ {σ : Type u_1} {R : Type u_3} [inst : CommSemiring R] {f : MvPowerSeries σ R} {m : ℕ},
MvPowerSeries.constantCoeff f ^ m = 0 →
∀ {d : σ →₀ ℕ} {n : ℕ}, m + Finsupp.degree d ≤ n → (MvPowerSeries.coeff d) (f ^ n) = 0Vanishing of coefficients of powers of multivariate power series when the constant coefficient is nilpotent [N. Bourbaki, Algebra II, Chapter 4, §4, n°2, proposition 3][bourbaki1981]
- Defined in
- Mathlib.RingTheory.MvPowerSeries.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
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Cites44
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- RingHom.idstatement · cited by 18,349
- Finsetproof · cited by 13,712
- AddCommMonoidproof · cited by 12,281
- CommSemiringstatement and proof · cited by 10,911
- LinearMapstatement · cited by 10,215
- RingHomstatement · cited by 10,189
- Finsuppstatement and proof · cited by 5,255
- Finset.sumproof · cited by 5,195
- mul_oneproof · cited by 3,885
- AddMonoidHomstatement · cited by 3,230
- add_zeroproof · cited by 2,707
Cited by2
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