nLab square-free element



The generalization of square-free integer and square-free polynomial to arbitrary unique factorization domains.


Let RR be a unique factorization domain. An element rRr \in R is square-free if for all irreducible elements xRx \in R, rr is not in the ideal x 2Rx^2 R.



Given a unique factorization domain RR such that for every irreducible element pRp \in R, the ideal pRp R is a maximal ideal, and a square-free element xRx \in R, the quotient ring R/xRR/x R is a reduced ring. Since for every non-zero element rRr \in R, R/xRR/x R is a prefield ring, R/xRR/x R is an integral domain and thus a field if and only if xx is an irreducible element in RR.

Square-free factorization

Given a non-zero element of a unique factorization domain rRr \in R, the square-free factorization or square-free decomposition of rr is a factorization of rr into powers of square-free elements r ir_i

r= i=0 nr i ir = \prod_{i = 0}^{n} r_i^i

where each of the elements r ir_i are pairwise coprime and not in the group of units.

See also

Last revised on January 22, 2023 at 20:45:31. See the history of this page for a list of all contributions to it.