nLab square-free number



A square-free number is a natural number nn such that for all prime numbers pp, p 2p^2 does not divide nn.


Given a square-free number nn with prime factors

n=p in = \prod p_i

the completion of the sequence of finite cyclic rings /n i\mathbb{Z}/n^i\mathbb{Z} is the ring n\mathbb{Z}_n, which is a subring of the profinite integers and the product of the p-adic integers of the prime factors of nn:

n p i\mathbb{Z}_n \coloneqq \prod \mathbb{Z}_{p_i}

Given a square-free integer nn, the integers modulo n /n\mathbb{Z}/n\mathbb{Z} is a reduced ring. Since for every integer nn the /n\mathbb{Z}/n\mathbb{Z} is a prefield ring, /n\mathbb{Z}/n\mathbb{Z} is an integral domain and thus a field if and only if nn is a prime number.

Given any odd prime number pp, the integer 2p2 p is square-free, and the ring /2p\mathbb{Z}/2 p\mathbb{Z} has idempotent elements of 00, 11, pp, and p+1p + 1. Every element of /2p\mathbb{Z}/2 p\mathbb{Z} could be written as a linear combination of pp and p+1p + 1.

See also

Last revised on January 22, 2023 at 22:20:16. See the history of this page for a list of all contributions to it.