Quote by joebohr
Ok, can we explore some example problems in which these extensions would come up?
Also, how would absolute Galois Groups fit into this?

The story is a bit too complicated to tell here, but one basic example comes from trying to answer the question: what are the primes that can be written in the form x^2 + ny^2 (with x, y in Z)? Whenever you write p=x^2+ny^2, this amounts to saying that p can be "factored" as p=(xsqrt(n)y)(x+sqrt(n)y) in Q(sqrt(n)). This kind of factorization can be encoded in the Galois group of Q(sqrt(n))/Q.
This is just one way field extensions show up in number theory.