Let i and j be integers such that gcd(i,n) = gcd(j,n). Let r be the ratio (1-yi)/(1-yj). Use synthetic division to expand the quotient, giving the following:
1 + yj + y2j + y3j + … + yk
The quotient terminates because there is some k such that jk = i mod n. Divide through by the common gcd, whence j/g becomes a unit mod n/g, and some k times j/g gives i/g.
The inverse, (1-yj)/(1-yi), is also a polynomial in y, hence r is invertible.
Some of these expressions produce old familiar units. When n = 5, (1-y2)/(1-y3) = 1+y3+y+y4, or -y2. Others are new, such as (1-y2)/(1-y) = 1+y.