Let $K/k$ be a [field extension](/page/Field%20Extension), let $n\in\mathbb N$ with $n\ge 1$, and let $A\in M_n(k)$. Let $\iota:k\hookrightarrow K$ be the inclusion map, and let $\iota_*:k[t]\hookrightarrow K[t]$ be the induced coefficientwise inclusion. If $\chi_{A,k}(t)\in k[t]$ denotes the [characteristic polynomial](/page/Characteristic%20Polynomial) of $A$ computed in $M_n(k)$, and $\chi_{A,K}(t)\in K[t]$ denotes the characteristic polynomial of the same matrix regarded as an element of $M_n(K)$, then

\begin{align*} \chi_{A,K}(t)=\iota_*(\chi_{A,k}(t)). \end{align*}