mathlib documentation

algebra.squarefree

Squarefree elements of monoids #

An element of a monoid is squarefree when it is not divisible by any squares except the squares of units.

Results about squarefree natural numbers are proved in data/nat/squarefree.

Main Definitions #

Main Results #

Tags #

squarefree, multiplicity

def squarefree {R : Type u_1} [monoid R] (r : R) :
Prop

An element of a monoid is squarefree if the only squares that divide it are the squares of units.

Equations
Instances for squarefree
@[simp]
theorem is_unit.squarefree {R : Type u_1} [comm_monoid R] {x : R} (h : is_unit x) :
@[simp]
theorem squarefree_one {R : Type u_1} [comm_monoid R] :
@[simp]
theorem not_squarefree_zero {R : Type u_1} [monoid_with_zero R] [nontrivial R] :
theorem squarefree.ne_zero {R : Type u_1} [monoid_with_zero R] [nontrivial R] {m : R} (hm : squarefree m) :
m 0
@[simp]
theorem irreducible.squarefree {R : Type u_1} [comm_monoid R] {x : R} (h : irreducible x) :
@[simp]
theorem prime.squarefree {R : Type u_1} [cancel_comm_monoid_with_zero R] {x : R} (h : prime x) :
theorem squarefree.of_mul_left {R : Type u_1} [comm_monoid R] {m n : R} (hmn : squarefree (m * n)) :
theorem squarefree.of_mul_right {R : Type u_1} [comm_monoid R] {m n : R} (hmn : squarefree (m * n)) :
theorem squarefree_of_dvd_of_squarefree {R : Type u_1} [comm_monoid R] {x y : R} (hdvd : x y) (hsq : squarefree y) :
theorem multiplicity.finite_prime_left {R : Type u_1} [cancel_comm_monoid_with_zero R] [wf_dvd_monoid R] {a b : R} (ha : prime a) (hb : b 0) :
theorem irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree {R : Type u_1} [comm_monoid_with_zero R] [wf_dvd_monoid R] (r : R) :
(∀ (x : R), irreducible x¬x * x r) (r = 0 ∀ (x : R), ¬irreducible x) squarefree r
theorem squarefree_iff_irreducible_sq_not_dvd_of_ne_zero {R : Type u_1} [comm_monoid_with_zero R] [wf_dvd_monoid R] {r : R} (hr : r 0) :
squarefree r ∀ (x : R), irreducible x¬x * x r
theorem squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible {R : Type u_1} [comm_monoid_with_zero R] [wf_dvd_monoid R] {r : R} (hr : ∃ (x : R), irreducible x) :
squarefree r ∀ (x : R), irreducible x¬x * x r