How is defined an invariant of a field? Given a certain field extension $L/K$, is it related with the Galois group ${\rm Gal}(L/K)$? In the case of number fields, which are the invariants associated to these extensions? For example, I remember that the relative discriminant is an invariant, but any other examples? Is it possible that the maximum power of an element $a$, i.e. the maximum $h$ such that $a=b^h$ for some element $b$, is again an invariant?
I know the question is very broad, so any bibliografy/reference suggestion would be great too!
Thanks in advance
I will try it with a short answer (to the title question), only giving two further invariants. Besides the discriminant also the ideal class group and its order, the class number are invariants, and the ring of integers of a number field in general. Furthermore Minkowski's bound, and the invariants involved there, i.e., the number of real and complex embeddings.