These notes from the Uni. Connecticut state about finite extensions of fields that "among elementary applications, the field trace can be used to show certain numbers are not in certain fields", but follow up discussion does not delve much into this concept.
Edit:
I got in touch with the author of the notes who kindly updated the document. There are referenced examples inside on Section 4 which show how the field trace can be used in such a way. Still it remains to be generalised for other cases, e.g. prime finite field examples or cases over binary extension fields:
How can the trace of an element $e \in L/K$, L over K, reflect participation (or not) in the field extension $L$? I come from an engineering background so can someone please explain in summary why this holds and provide an example? Thanks!