Alladi Ramakrishnan Hall

Consequences of Failure of Chinese Remainder Theorem in the Ring of Polynomials over Integers

Kamalakshya Mahatab and Kannappan Sampath

IMSc, ISI Bangalore

We investigate questions arising from non-surjectivity of the map:

\[\mathbf{Z}[X]/(X^n - 1) \to \bigoplus_{d | n} \mathbf{Z}[X]/(\Phi_d(X))\]

where $\Phi_d$ is the $d$th cyclotomic polynomial. We determine the elementary divisors of this map, thus describing its cokernel. We will describe a combinatorial basis for the cokernel. We'll also describe a combinatorial decomposition for $\mathbf{Z}[X]/(X^n - 1)$. This method generalizes to quotients of polynomial rings over PID by monic polynomials and group ring of finite abelian groups.