This talk and its sequels will give a survey of results on the non-existence of certain Galois extensions of the rational number field Q with prescribed ramification. The talks are mostly based on two papers: one by Sharon Brueggeman (The non-existence of certain Galois extensions unramified outside 5) and the other by Hyunsuk Moon and Yuichiro Taguchi (a refinement of Tate's discriminant bound and non-existence theorems for mod p Galois representations). I will start from Serre's conjecture, which says that every continuous irreducible odd mod p Galois representation comes from a modular form. Motivated by Serre's conjecture, I will show the non-existence of certain continuous irreducible mod p representations of degree 2 of the absolute Galois group of the rational number field. In the first two talks I will discribe the work done by Tate and Brueggeman about the small p case (p=2,5), and a generalization to other primes. Afterwards I will discuss Moon and Taguchi's refinement of the Tate's discriminant bound to get the case when p<=31 with small Serre weight k.