To amplify the points made by Laurent Berger, the literature I've seen (dating from around 1950) always specifies perfect fields; so I believe it was understood very early that Jordan-type decomposition breaks down for imperfect fields. I'm not sure whether this is discussed in the modern textbooks on theoretical linear algebra such as those by Curtis and Hoffman-Kunze. (In basic linear algebra it's rare to mention imperfect fields, though this becomes a real issue in Lie theory.)
Beyond the Jordan normal form for a matrix (originally developed over a field of characteristic 0 containing all the eigenvalues), the work of Chevalley has been essential for the more flexible notion of "Jordan decomposition" and related matrix polynomials over a perfect field not containing the eigenvalues. Of course he was motivated especially by the theory of linear algebraic groups, but even for computational linear algebra his viewpoint is historically important and justifies the term Jordan-Chevalley decomposition Much of the history has been written down in a joint paper by Danielle Couty and colleagues: see the arXiv preprint here.
