Groups Of Units Of Z_P[X] Modulo F(X)

Abstract

The set $\mathbb{Z}_p[x]$ consists of all polynomials with coefficients in the field $\mathbb{Z}_p$, where $p$ is prime. If a polynomial $f(x)$ is irreducible over $\mathbb{Z}_p$ then $\frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}$ is a field. If $f(x)$ is a reducible polynomial, then every non-zero element in $\frac{\mathbb{Z}_p[x] }{\langle f(x) \rangle}$ is either a zero-divisor or a unit. If we exclude the zero-divisors and zero, we have a finite Abelian group under multiplication denoted by $U \Big( \frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}\Big) $. Since every finite Abelian group is a direct product of cyclic groups of prime-power order, we can find the isomorphism class for $U \Big( \frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}\Big) $. We investigate the structure of $U \Big( \frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}\Big) $ for a prime $p$ and various $f(x)$. We conclude with some result on the structure of a certain family of subgroups of $U \Big( \frac{\mathbb{Z}_p[x]}{\langle f(x)\rangle} \Big)$.