Theorems · Theorem · commutative algebra
Irreducible.isUnit_or_isUnit
∀ {M : Type u_1} [inst : Monoid M] {p : M}, Irreducible p → ∀ ⦃a b : M⦄, p = a * b → IsUnit a ∨ IsUnit bIf an irreducible element factors, then one factor is a unit.
- Defined in
- Mathlib.Algebra.Group.Irreducible.Defs
- Cited by
- 28 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
- Assumes
- Monoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Monoidstatement and proof · cited by 3,887
- IsUnitstatement · cited by 1,602
- Irreduciblestatement and proof · cited by 496
Cited by28
Results whose statement or proof uses this declaration.
- Nat.Prime.eq_one_or_self_of_dvdproof · cited by 11
- Associated.irreducibleproof · cited by 10
- minpoly.eq_of_irreducible_of_monicproof · cited by 9
- Polynomial.degree_eq_one_of_irreducible_of_rootproof · cited by 7
- Irreducible.squarefreeproof · cited by 6
- irreducible_iffproof · cited by 5
- Polynomial.IsPrimitive.irreducible_iff_irreducible_map_fraction_mapproof · cited by 4
- Irreducible.dvd_iffproof · cited by 4
- pow_ne_of_irreducible_X_pow_sub_Cproof · cited by 4
- Irreducible.of_mapproof · cited by 4
- Polynomial.irreducible_of_monicproof · cited by 3
- Polynomial.IsPrimitive.irreducible_of_irreducible_map_of_injectiveproof · cited by 2