Factor rings of the form Zp[x]/, with p prime and f(x) irreducible in Zp[x], form a field, with cyclic multiplicative group structure. When f(x) is reducible in Zp[x] this factor ring is no longer a field, nor even an integral domain, and the structure of its group of units is no longer cyclic. In this paper we develop concise formulas for determining the cyclic group decomposition of the multiplicative group of units for Zp[x]/ that is only dependent on the multiplicities and degrees of the irreducible factors of f(x), and p.
Gerhold, Erika; Ferralli, Jennifer; and Jachowski, Jason
"The Multiplicative Structure of the Group of Units of ℤₚ[𝑥]/〈𝑓(𝑥)〉 where 𝑓(𝑥) is Reducible,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 16:
1, Article 5.
Available at: https://scholar.rose-hulman.edu/rhumj/vol16/iss1/5