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For fixed odd modulus, the Jacobi symbol is a homomorphism on the unit group

Statement

Fix an odd positive integer $n$ . The assignment $χ_{n} ([a]_{n}) = (\frac{a}{n})$ is a group homomorphism $(Z / n)^{\times} \to {\pm 1}$ .

Here ${\pm 1}$ is the two-element multiplicative group, except that the image is the one-element subgroup ${1}$ when the character is trivial.

Facts & Assumptions

Given: An odd positive integer $n$ and unit classes $[a]_{n}, [b]_{n} \in (Z / n)^{\times}$ .

[L1]

The Jacobi symbol belongs to ${- 1, 0, 1}$ , depends only on $a (m o d n)$ , and is zero exactly when $\gcd (a, n) > 1$ (The Jacobi symbol is well defined on numerator residue classes).

[L2]

For odd positive $n$ , $(\frac{a b}{n}) = (\frac{a}{n}) (\frac{b}{n})$ (The Jacobi symbol is multiplicative in numerator and denominator).

[L3]

The unit group $(Z / n)^{\times}$ consists of the invertible residue classes modulo $n$ under multiplication (The unit group $(Z / n)^{\times}$ and Euler's totient $φ (n) = ∣ (Z / n)^{\times} ∣$ for $n \geq 1$ ).

[L4]

The class $[a]_{n}$ is a unit if and only if $\gcd (a, n) = 1$ (For $n \geq 1$ , $[a]_{n}$ is a unit if and only if $\gcd (a, n) = 1$ ).

[L5]

A group homomorphism $f : G \to H$ is a function satisfying $f (x y) = f (x) f (y)$ for all $x, y \in G$ (Monoid homomorphism and group homomorphism).

Proof

technique · direct

1.1

By [L1], the value $(\frac{a}{n})$ depends only on the residue class. By [L3] and [L4], a unit class has $\gcd (a, n) = 1$ , so [L1] rules out the value zero; hence $χ_{n}$ is a well-defined function from $(Z / n)^{\times}$ to ${\pm 1}$ .

L1L3L4

2.1

For unit classes $[a]_{n}$ and $[b]_{n}$ , [L2] gives $χ_{n} ([a]_{n} [b]_{n}) = χ_{n} ([a b]_{n}) = χ_{n} ([a]_{n}) χ_{n} ([b]_{n})$ , which is the condition in [L5]. Thus $χ_{n}$ is a group homomorphism, including when $n = 1$ and the unit group has one element.

step 1.1L2L5∎

Depends on

Used by

A unit square modulo an odd integer has Jacobi symbol one Proposition
The kernel of the Jacobi map and the subgroup of unit squares Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 82 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

V. Shoup, A Computational Introduction to Number Theory and Algebra, 2nd ed., §12.2 (standard reference, not scraped)