Theorems · Theorem · field theory
Polynomial.isUnit_iff
∀ {R : Type u} [inst : Semiring R] [NoZeroDivisors R] {p : Polynomial R}, IsUnit p ↔ ∃ r, IsUnit r ∧ Polynomial.C r = pCharacterization of a unit of a polynomial ring over an integral domain R.
See Polynomial.isUnit_iff_coeff_isUnit_isNilpotent when R is a commutative ring.
- Defined in
- Mathlib.Algebra.Polynomial.Degree.Units
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringNoZeroDivisors
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Semiringstatement and proof · cited by 13,802
- RingHomstatement · cited by 10,189
- Polynomialstatement and proof · cited by 5,681
- IsUnitstatement and proof · cited by 1,602
- Polynomial.Cstatement and proof · cited by 1,598
- Polynomial.coeffproof · cited by 1,045
- NoZeroDivisorsstatement and proof · cited by 545
- Polynomial.eq_C_of_natDegree_eq_zeroproof · cited by 22
- Polynomial.isUnit_Cproof · cited by 19
- Polynomial.natDegree_eq_zero_of_isUnitproof · cited by 5
Cited by10
Results whose statement or proof uses this declaration.
- Polynomial.isUnit_or_eq_zero_of_separable_expandproof · cited by 2
- Polynomial.IsPrimitive.isUnit_iff_isUnit_map_of_injectiveproof · cited by 2
- Irreducible.isPrimitiveproof · cited by 2
- Polynomial.isUnit_or_eq_zero_of_isUnit_integerNormalization_primPartproof · cited by 1
- Polynomial.resultant_dvd_leadingCoeff_powproof · cited by 1
- Polynomial.natSepDegree_mul_eq_iffproof · cited by 1
- Polynomial.irreducible_of_degree_eq_one_of_isRelPrime_coeffproof · cited by 1
- Polynomial.unique_separable_of_irreducibleproof · cited by 0
- Polynomial.Monic.irreducible_of_irreducible_map_of_isPrime_nilradicalproof · cited by 0
- Polynomial.coeff_mem_radical_span_coeff_of_dvdproof · cited by 0