Direct Construction of Nontrivial Square Roots of Unity Modulo N from Known Factorization

This note studies explicit constructions of nontrivial square roots of unity modulo composite integers N = pq, where p and q are distinct odd primes. Assuming the factorization of N is known, we derive a direct closed-form expression for a nontrivial square root of unity modulo N, namely (q-p)/(q+p) mod N, and prove its equivalence to an exponentiation-based construction using the Carmichael function λ(N). Both constructions are further interpreted through the algebraic framework of primitive idempotents arising from the Chinese Remainder Theorem decomposition of ℤ/Nℤ. The paper provides a concise unified perspective connecting CRT structure, involutions, idempotents, and factorization-related properties of RSA moduli.