Theorems · Theorem · field theory
IsAlgClosed.exists_root
∀ {k : Type u} [inst : Field k] [IsAlgClosed k] (p : Polynomial k), p.degree ≠ 0 → ∃ x, p.IsRoot xIf k is algebraically closed, then every nonconstant polynomial has a root.
- Defined in
- Mathlib.FieldTheory.IsAlgClosed.Basic
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldIsAlgClosed
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- WithBotstatement · cited by 1,498
- Polynomial.degreestatement and proof · cited by 643
- Polynomial.IsRootstatement · cited by 152
- IsAlgClosedstatement and proof · cited by 150
- IsAlgClosed.splitsproof · cited by 29
- Polynomial.Splits.exists_eval_eq_zeroproof · cited by 14
Cited by7
Results whose statement or proof uses this declaration.
- IsPrimitiveRoot.sub_one_norm_eq_eval_cyclotomicproof · cited by 3
- IsAlgClosed.exists_eval₂_eq_zero_of_injectiveproof · cited by 2
- IsAlgClosed.exists_pow_nat_eqproof · cited by 2
- IsAlgClosed.roots_eq_zero_iffproof · cited by 1
- WittVector.RecursionMain.root_existsproof · cited by 1
- Polynomial.isCoprime_iff_aeval_ne_zero_of_isAlgClosedproof · cited by 0
- IsAlgClosed.of_ringEquivproof · cited by 0