Example

Let $K$ be a field, $a, (1 \leq i \leq m)$ distinct elements of $K$ and, for each $i$, let $g_i$ be a polynomial in $K[X]$; the principal ideal $(X - a_i) = m_i$ is maximal in $K[X]$, hence, for every system $(n_i)_{1 \leq i \leq m}$ of $m$ integers $\geq 1$, the ideals $m_i^{n_i}$ are relatively prime in pairs. Then it follows from Proposition 5 that there exists a polynomial $f \in K[X]$ such that $f(X) \equiv g_i(X) \pmod{(X - a_i)^{n_i}}$ for $1 \leq i \leq m$, the difference of two such polynomials being divisible by $\omega(X) = \prod_{i=1}^m (X - a_i)^{n_i}$. If all the $n_i$ are taken equal to 1, we find the problem is solved explicitly by Lagrange's interpolation formula (Algebra, Chapter IV, § 2, no. 4).