
Re: What exactly is a socalled squarefree integer?
Posted:
Apr 28, 1999 3:49 PM


Bill Taylor <mathwft@math.canterbury.ac.nz> wrote:  Bob Silverman <bobs@rsa.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 squarefree 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 nonUFDs. 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.
Bill Dubuque

