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Topic: What exactly is a so-called squarefree integer?
Replies: 11   Last Post: May 17, 1999 7:10 PM

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Bill Dubuque

Posts: 156
Registered: 12/6/04
Re: What exactly is a so-called squarefree integer?
Posted: Apr 28, 1999 3:49 PM
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Bill Taylor <> wrote:
| Bob Silverman <> 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.

-Bill Dubuque

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