The kernel of the Jacobi map and the subgroup of unit squares

Given: An odd positive integer $n={\prod }_{i<r}{p}_i^{e_i}$ in canonical prime factorisation and its unit group $U_n$.

The assignment ${\chi }_n([a{]}_n)=\left(\frac{a}{n}\right)$ is a group homomorphism $U_n\to \{\pm 1\}$ (For fixed odd modulus, the Jacobi symbol is a homomorphism on the unit group).

Every unit square modulo an odd positive integer has Jacobi symbol one (A unit square modulo an odd integer has Jacobi symbol one).

For odd $n={\prod }_{i<r}{p}_i^{e_i}$, the soluble unit congruence $x^2\equiv a(\mathrm{m}\mathrm{o}\mathrm{d}n)$ has exactly $2^r$ roots (The number of square roots of a unit modulo $n$ is the product of the local counts).

For every odd prime $p$ and $k\ge 1$, the group $(\mathbb{Z}\mathrm{/}{p}^k{)}^{\times }$ is cyclic of even order $p^{k-1}(p-1)$ (For every odd prime $p$ and $k\ge 1$, $(\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z}{)}^{\times }$ is cyclic of order $p^{k-1}(p-1)$).

The Chinese remainder map gives $U_n\cong {\prod }_{i<r}(\mathbb{Z}\mathrm{/}{p}_i^{e_i}{)}^{\times }$, including the empty factorisation (For pairwise coprime positive moduli, the Chinese remainder bijection restricts to an isomorphism of unit groups).

For a group homomorphism $f:G\to H$, one has $G\mathrm{/}\ker f\cong \mathrm{im}f$ (First isomorphism theorem for groups: $G\mathrm{/}\ker f\cong \mathrm{im}f$).

If $G$ is finite and $H\le G$, then $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ (Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$).

For a finite-index subgroup, $[G:H]=\mathrm{\mid }G\mathrm{/}H\mathrm{\mid }$ (The coset set $G\mathrm{/}H$ and the index $[G:H]$ of a subgroup).

If $K\le H\le G$ and $G$ is finite, then $[G:K]=[G:H][H:K]$ (For $K\le H\le G$ with $G$ finite, $[G:K]=[G:H][H:K]$).

For an odd prime $p$, a unit is a square modulo $p^k$ if and only if its Legendre symbol modulo $p$ is one (Unit square criterion and root count modulo odd prime powers).

For a homomorphism $f:G\to H$, $\ker f=\{g\in G:f(g)=e_H\}$ and $\mathrm{im}f=\{f(g):g\in G\}$ (The kernel and image of a group homomorphism).

Canonical prime-factor exponents are determined by the integer (For $n\ge 1$ and any injective list $p:r\to \mathbb{Z}$ of primes containing every prime divisor of $n$, one has $n={\prod }_{i<r}{p}_i^{v_{p_i}(n)}$; the exponents are determined by $n$, and $v_q(n)=0$ for every prime $q$ outside the list).

For a prime $p$ and nonzero integers $u,v$, $v_p(uv)=v_p(u)+v_p(v)$ ($v_p(ab)=v_p(a)+v_p(b)$ for nonzero integers $a,b$, and $v_p(a+b)\ge \min \{v_p(a),v_p(b)\}$ whenever $a$, $b$ and $a+b$ are all nonzero).

The Jacobi symbol is the product of the prime Legendre symbols with their canonical multiplicities (The Jacobi symbol, with its zero value and empty-product convention).