A good reference to the Gauss result on the structure of the multiplicative group of a residue ring

Question

I need a good reference (desirably some textbook in Number Theory) to the following known result, attributed to Gauss in Wikipedia.

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!

Answer 1

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.

Answer 2

Another reference is Proposition 2.1.24, page 22, in Henri Cohen's "Number Theory – Volume I: Tools and Diophantine Equation".