In mathematics, a unit in a ring R is an element u such that there is v in R with
uv = vu = 1R.
That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of units of R.
The orbits of U(R) acting on R by multiplication are called sets of associates; in other words there is an equivalence relation on R called associatedness such that
r ~ s
means that there is a unit u with r = us. For example in the ring Z of integers n and −n are associates.
In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a "generator" of the group) such that all elements of the group are powers of a.
All subgroups and factor groups of cyclic groups are cyclic.
The Galois group of every finite field extension of a finite field is finite and cyclic; conversely, given a finite field F and a finite cyclic group G, there is a finite field extension of F whose Galois group is G.
Feeds |
Contact