Bill Taylor <firstname.lastname@example.org> wrote: | Bob Silverman <email@example.com> writes: | > | > square free means that every prime dividing the number | > does so to the first power. | | Neat enough in dealing with the anomalous case of "1". | | When I taught this I liked to point out the amusing fact that | 1 was BOTH square-free AND a perfect square! Cutely enraging! | =========== ============== | Needed, of course, to ensure that every natural number has | a unique decomposition into a product of one of each type.
Squarefree simply means free of (nonunit) square factors, e.g. a polynomial p in k[x] is squarefree if gcd(p,p')=1.
Unlike this (global) definition, the quoted (local) definition may break down in non-UFDs. Even if one replaces "prime" with "irreducible" (to handle inequivalent irreducible factorizations like p^2 = q r ) one still has the problem that a square factor might have no factorization into irreducibles; e.g. in the ring I of all algebraic integers there are no irreducibles but also no squarefrees since w = sqrt(w)^2 for every element w in I.