Exercise 3.1.5 (Solve $x^2 \equiv a \pmod {11}$ and $x^2 \equiv a \pmod {11^2}$)

Proof. With $p=11$,

Since $a^2\equiv {(p-a)}^2(\mathit{mod}p)$, it is sufficient to list all squares $a^2\mathit{mod}p$ for $a\in [[1,(p-1)∕2]]$. So the list of all quadratic residues of $11$ are

The results are given in this array:

To give an example, with $a=5$, $x^2\equiv 5(\mathit{mod}11)$ has solutions $x\equiv 4,7(\mathit{mod}11)$ by the first array.

To solve $x^2\equiv 5(\mathit{mod}11^2)$, we write $x=4+11k$, where $k$ is an integer (or $x=7+11k$). In the first case, $16+8\cdot 11k\equiv 5(\mathit{mod}11^2)$, thus $8k\equiv -1(\mathit{mod}11)$, so $k\equiv 4(\mathit{mod}11)$, $k=4+11\lambda$, and $x\equiv 4+11(4+11\lambda )(\mathit{mod}11^2)$, thus $x\equiv 48(\mathit{mod}11^2)$. Similarly, in the second case, $x\equiv 73(\mathit{mod}11^2)$. □