When I taught courses on algebra giving a first exposition to Galois theory I usually included some discussion of classical results showing the impossibility of constructing certain points with ruler and compass, such a the cube root of $2$ (Delian problem) and the trisection of a (generic) angle.
However, sometimes I think maybe this is not that good an allocation of time. It took me quite some time to recall and/or to introduce in precise terms what it means to construct something, which feels a bit like an isolated subject in such a course. Likewise, the results, while historically important and interesting, do not seem of much use later on. Since as said I did include the subject, needless to say, I can also see some merit in it. Yet, I never quite managed to make up my mind.
Thus, I would like further input on this subject:
What are reasons for or against teaching the impossibility of certain geometric constructions in a course on algebra that covers the basics of Galois theory?