Let $K$ be a complete local field with discrete valuation, and let $L/K$ be a finite Galois extension. Use $G=Gal(L/K)$ to denote the Galois group.
In Serre's book 'local fields', chapter 6, a linear representation of $G$, called Artin representation, is defined by identifying its character. Explicit description of such representations is discussed in the case of algebraic curves in section 4 of that chapter.
But I'm more interested in the case when $L,K$ are $p$-adic fields, and I wanna know the following questions:
(1) How to construct such representations explicitly in $p$-adic fields case?
(2) What's its role in ramification theory, or more general in (local) class field theory?
(3) What other properties do we know about such representations? Like when will it be irreducible?
Thank you very much for any answer or references for any of the above questions.