Unique factorization and normalization #
Main definitions #
UniqueFactorizationMonoid.normalizedFactors
: choose a multiset of prime factors that are unique by normalizing them.UniqueFactorizationMonoid.normalizationMonoid
: choose a way of normalizing the elements of a UFM
Noncomputably determines the multiset of prime factors.
Equations
Instances For
An arbitrary choice of factors of x : M
is exactly the (unique) normalized set of factors,
if M
has a trivial group of units.
Alias of UniqueFactorizationMonoid.zero_notMem_normalizedFactors
.
Relatively prime elements have disjoint prime factors (as multisets).
The multiset of normalized factors of x
is nil if and only if x
is a unit.
The converse is true without the nonzero assumption, see normalizedFactors_of_isUnit
.
If x
is a unit, then the multiset of normalized factors of x
is nil.
The converse is true with a nonzero assumption, see normalizedFactors_eq_zero_iff
.
If the monoid equiv f : α ≃* β
commutes with normalize
then, for a : α
, it yields a
bijection between the normalizedFactors
of a
and of f a
.
Equations
Instances For
Noncomputably defines a normalizationMonoid
structure on a UniqueFactorizationMonoid
.
Equations
- One or more equations did not get rendered due to their size.