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For pairwise coprime positive moduli, the Chinese remainder bijection restricts to an isomorphism of unit groups
Statement
Let be pairwise coprime positive integers and . The Chinese remainder map restricts to a group isomorphism
For the empty list this identifies the two one-element groups.
Facts & Assumptions
Given: The stated finite pairwise-coprime list and its product .
The Chinese remainder map is a bijection preserving multiplication and identity, including for the empty list (Chinese remainder theorem for a finite pairwise-coprime list: simultaneous residues determine one class modulo the product, and the resulting bijection preserves addition and multiplication).
For , the invertible classes of form a group under multiplication (The unit group and Euler's totient for ).
A bijective group homomorphism is an isomorphism (Group isomorphisms, automorphisms and the set ).
Proof
By [L1], the CRT map preserves multiplication and identity. If a class has an inverse, its image has the coordinatewise image of that inverse.
Conversely, if every coordinate is a unit, take the tuple of coordinatewise inverses and use surjectivity in [L1] to lift it; multiplicativity shows that the lift is an inverse of the original class.
Steps 1.1 and 1.2 show that [L1] restricts to a bijective homomorphism between the displayed unit groups.
It is therefore an isomorphism by [L3], and [L1] supplies the empty-list case.
Depends on
- Chinese remainder theorem for a finite pairwise-coprime list: simultaneous residues determine one class modulo the product, and the resulting bijection preserves addition and multiplication
- The unit group $(\mathbb{Z}/n)^\times$ and Euler's totient $\varphi(n)=\lvert(\mathbb{Z}/n)^\times\rvert$ for $n\ge1$
- Group isomorphisms, automorphisms and the set $\operatorname{Aut}(G)$
Used by
- The number of square roots of a unit modulo n is the product of the local counts Corollary
- For odd n, primitive-root existence is equivalent for n and 2n Lemma
- A unit is a square modulo n exactly when it is a square at every prime-power factor Theorem
- The kernel of the Jacobi map and the subgroup of unit squares Theorem
- The unit group modulo n is the product of its odd-prime cyclic factors and its explicit 2-power factor Theorem
Dependency tree · two levels
20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Peter Hackman, Elementary Number Theory, §C.V (standard reference, not scraped)