Overleg:Kwadraatvrij geheel getal

Negatieve d

Verplaatst naar kwadratisch lichaam Madyno (overleg) 16 nov 2021 17:10 (CET)

vertalen

Er stond als commentaar dat de volgende tekst niet mee moet worden vertaald:

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)