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Let $\mathbb{H}$ be a field with$\mathbb{Q}\subset\mathbb{H}\subset\mathbb{C}$, and let $p(\lambda)$ be apolynomial in $\mathbb{H}[\lambda]$, and let $A\in\mathbb{H}^{n\times n}$ benonderogatory. In this paper we consider the problem of finding a solution$X\in\mathbb{H}^{n\times n}$ to $p(X)=A$. A necessary condition for this to bepossible is already known from a paper by M.P. Drazin. Under an additionalcondition we provide an explicit construction of such solutions. Thesimilarities and differences with the derogatory case will be discussed aswell.  One of the tools needed in the paper is a new canonical form, which may be ofindependent interest. It combines elements of the rational canonical form withelements of the Jordan canonical form.

Solutions of the matrix equation $p(X)=A$, with polynomial function $p(\lambda)$ over field extensions of $\mathbb{Q}$