Ramification theory and Artin Represenration

For every finite Galois extension of a local separable field we can constuct ramification groups $G_{i}$. It is clear from definition that these groups behave well under taking subgroups of the Galois group $G$, but the same is not true for taking quotients of $G$. I will try to explain how we can define upper-numbering $G^{i}$ which will behave well under taking quotients. After this, I will give a construction of an Artin character. A non-obvious fact is that it is a character of a linear representation. I will give a plan of the proof of this fact and at the end we will discuss some application of this theory to number fields.