What is the structure of the multiplicative group of the ring $\mathbb{Z}_{p^k } $?

Question

I want to find the automorphism group of $\mathbb{Z}_n $ for a general integer $n$, and I think that it is related to the structure of the multiplicative group of $\mathbb{Z}_n $, however I can't even determine the case when $n=p^k $. What is the structure of the multiplicative group $\mathbb{Z}_{p^k }^{\times } $?

Answer 1

I have found the answer in Ireland&Rosen's book "A Classical Introduction to Modern Number Theory" (GTM Vol.84). If $n=2^a p_1^{a_1 } \dotsb p_l^{a_l } $, then $$U(\mathbb{Z} /n\mathbb{Z} )\cong U(\mathbb{Z} /2^a \mathbb{Z} )\times\dotsb\times U(\mathbb{Z} /p_l^{a_l } \mathbb{Z} )$$ where $U(\mathbb{Z} /p_i^{a_i } \mathbb{Z} )$ is a cyclic group, and $U(\mathbb{Z} /2^a \mathbb{Z} )$ is a cyclic group for $a=1$ or $2$ but a product of two cyclic groups with one of order $2$ and the other of order $2^{a-2 } $. Its proof is in chapter $4$.