The trace and norm are very useful as maps from a field extension to the base field since they are multiplicative/additive and have a lot of other nice properties. They can be defined as two of the coefficients of the irreducible polynomial of the element over the base field.
These are the only coefficients that are guaranteed to exist for any element of non trivial degree but other coefficients can be defined for higher degree elements. However, I have never seen any of them used in either Galois theory or algebraic number theory.
The obvious reason for this is that the other coefficients are neither multiplicative not additive but they still seem fairly important and it would be surprising if they were never useful.
What are some applications of the other coefficient(if any) and is there any other reason why they appear so infrequently?