I am looking for some motivation for the definitions of trace, norm and discriminant (in the context of finite field extensions). For example (but not limited to) any interesting theorems proved using techniques involving the mentioned above, or intuition on why and how they were defined in the first place.
Thanks.
Part of the motivation, at least for the trace and the norm, is that they are functions from what you want to study (the number field) to what you already know about (the rationals), and thus they may enable you to reduce questions you can't immediately answer to ones that you can. Just to give one tiny example, to prove $\alpha=1+\sqrt{-5}$ is irreducible in the ring $\{{a+b\sqrt{-5}:a,b{\rm\ in\ }{\bf Z}\}}$ of integers of ${\bf Q}(\sqrt{-5})$, you may assume $\alpha=\beta\gamma$, and take norms: $6=N(\beta)N(\gamma)$. Since the equations $a^2+5b^2=2$ and $a^2+5b^2=3$ have no integer solutions, you must have $N(\beta)=1$ or $N(\gamma)=1$, from which you deduce that one of the numbers $\beta,\gamma$ is a unit, so $\alpha$ is irreducible.