Why does invertibility of $2$ in $\mathbb{Z}_p$ guarantee that every element is a square?

Question

In Serre's arithemetic he claims at the top of page 18 that because $2$ is invertible in $\mathbb{Z}_p$ we have that every element in the multiplicative group $U_1 = 1 + p\cdot\mathbb{Z}_p$ is a square. Why is this true? The only step I can show is the invertibility of $2$ in $\mathbb{Z}_p$. Using the power series expansion $$ \frac{1}{1 + p} = 1 - p + p^2 - p^3 + p^4 - \cdots $$ we can see that $$ \frac{1}{2} = \frac{p+1}{2}\cdot (1 - p + p^2 - p^3 + p^4 - \cdots) \in \mathbb{Z}_p $$

Answer 1

$U_1=1+p\mathbb{Z}_p$ is a group under multiplication, which is isomorphic to the additive group $\mathbb{Z}_p$. The fact that every element of $U_1$ is a square (for $p>2$) then follows from the fact that for every $x\in\mathbb{Z}_p$ there is some $y\in\mathbb{Z}_p$ such that $x=2y$, and this follows from the fact that $2$ is invertible in $\mathbb{Z}_p$.