Explicit constructions of extensions of complete fields of characteristic $0$

This survey article is devoted to $p$-extensions of complete discrete valuation fields of mixed characteristic where $p$ is the characteristic of the residue field. It is known that any totally ramified Galois extension with a non-maximal ramification jump can be determined by an Artin-Schreier equation, and the upper bound for the ramification jump corresponds to the lower bound of the valuation in the right-hand side of the equation. The problem of construction of extensions with arbitrary Galois groups is not solved.