I need a good reference (desirably some textbook in Number Theory) to the following known result, attributed to Gauss in [Wikipedia](https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n).



> 
> **Theorem (Gauss).** Let $p$ be a prime number, $k\in\mathbb N$ and $\mathbb Z\_{p^k}^\times$ be the multiplicative group of invertible elements of the residue ring $\mathbb Z\_{p^k}:=\mathbb Z/p^k\mathbb Z$.
> 
> 
> 1. If $p$ is odd, then the group $\mathbb Z\_{p^k}^\times$ is cyclic.
> 2. If $p=2$ and $k\ge 3$, then the element $-1+2^k\mathbb Z$ generates a $2$-element subgroup $C\_2\subset\mathbb Z\_{2^k}^\times$ and the element $5+2^k\mathbb Z$ generates a cyclic subgroup $C\_{2^{k-2}}\subset \mathbb Z\_{2^k}^\times$ of order $2^{k-2}$ such that $\mathbb Z\_{p^k}^\times=C\_2\oplus C\_{2^{k-2}}$.
> 
> 
> 


Please help!



A good reference to the Gauss result on the structure of the multiplicative group of a residue ring
This is proved in detail in Chapter 4 (a short one) of "A classical introduction to modern number theory" by Ireland and Rosen. Specifically see Theorems 2 and 2' in page 43 (second edition) and their proofs.


