Theorems · Theorem · commutative algebra
squarefree_iff_no_irreducibles
∀ {R : Type u_1} [inst : CommMonoidWithZero R] [WfDvdMonoid R] {x : R},
x ≠ 0 → (Squarefree x ↔ ∀ (p : R), Irreducible p → ¬p * p ∣ x)- Defined in
- Mathlib.Algebra.Squarefree.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 16 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- IsUnitproof · cited by 1,602
- CommMonoidWithZerostatement and proof · cited by 913
- Irreduciblestatement and proof · cited by 496
- Dvd.dvd.transproof · cited by 148
- Squarefreestatement and proof · cited by 112
- by_contraproof · cited by 60
- dvd_mul_leftproof · cited by 47
- Irreducible.not_isUnitproof · cited by 42
- WfDvdMonoidstatement and proof · cited by 37
- ne_zero_of_dvd_ne_zeroproof · cited by 26
- mul_dvd_mulproof · cited by 24
- WfDvdMonoid.exists_irreducible_factorproof · cited by 12
Cited by1
Results whose statement or proof uses this declaration.
- irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefreeproof · cited by 2