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Jan21-12, 08:01 PM
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Quote Quote by joebohr View Post
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=(x-sqrt(-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.