Overleg:Kwadraatvrij geheel getal
Negatieve d
bewerkenVerplaatst naar kwadratisch lichaam Madyno (overleg) 16 nov 2021 17:10 (CET)
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The positive integer n is square-free if and only if all abelian groups of order n are isomorphic, which is the case if and only if all of them are cyclic. This follows from the classification of finitely generated abelian groups.
The integer n is square-free iff the factor ring Z / nZ (see modular arithmetic) is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form Z / kZ is a field if and only if k is a prime.
For every positive integer n, the set of all positive divisors of n becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice. It is a Boolean algebra if and only if n is square-free.
Given the positive integer n, define the radical of the integer n by
- m = rad(n),
equal to the product of the prime numbers p dividing n. Then the square-free n are exactly the solutions of n = rad(n). ChristiaanPR (overleg) 20 nov 2023 12:22 (CET)