Let \(k\) be a field and let \(A\in\Mat_n(k)\) have characteristic polynomial that splits over \(k\), so that \(A\) is similar to its Jordan canonical form \(\JCF(A)\). Then the elementary divisors of \(A\), the prime-power factors in the decomposition of the \(k[x]\)-module \(k^n\) with \(x\) acting as \(A\), are the minimal polynomials \((x-\lambda)^s\) of the Jordan blocks \(J_s(\lambda)\) of \(\JCF(A)\), listed with multiplicity.
A Jordan block \(J_s(\lambda)\) is the matrix of multiplication by \(x\) on \(k[x]/((x-\lambda)^s)\) in the basis \((x-\lambda)^{s-1},\ldots,(x-\lambda),1\), and its minimal polynomial is \((x-\lambda)^s\). So the block decomposition of \(\JCF(A)\) is a decomposition \(k^n\cong\bigoplus k[x]/((x-\lambda)^s)\), one summand for each block, and the elementary divisors are unique up to order.