Theorems · Theorem · functional analysis
NormedAlgebra.Real.exists_isMonicOfDegree_two_and_aeval_eq_zero
∀ {F : Type u_1} [inst : NormedRing F] [inst_1 : NormedAlgebra ℝ F] [NormOneClass F] [NormMulClass F] (x : F),
∃ p, p.IsMonicOfDegree 2 ∧ (Polynomial.aeval x) p = 0If F is a normed ℝ-algebra with a multiplicative norm (and such that ‖1‖ = 1),
e.g., a normed division ring, then every x : F is the root of a monic quadratic polynomial
with real coefficients.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 301 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites42
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Realstatement and proof · cited by 25,697
- Polynomialstatement · cited by 5,681
- Norm.normproof · cited by 5,413
- Algebra.algebraMapproof · cited by 4,706
- Set.univproof · cited by 3,945
- Filter.Tendstoproof · cited by 3,814
- AlgHomstatement · cited by 3,236
- Filter.atTopproof · cited by 2,405
- Polynomial.Xproof · cited by 1,639
- Polynomial.Cproof · cited by 1,598
- NormedAlgebrastatement and proof · cited by 1,165
Cited by1
Results whose statement or proof uses this declaration.
- NormedAlgebra.Real.nonempty_algEquiv_orproof · cited by 0