TITLE: Why going to number fields in number field sieve help beat quadratic sieve?
QUESTION [11 upvotes]: To factor an $n$ bit integer number field sieve roughly takes $$e^{c{(\ln\ln n)^{\frac23}}({\ln n})^{\frac13}}$$ time while quadratic sieve takes  $$e^{c{(\ln\ln n)^{\frac12}}({\ln n})^{\frac12}}$$ time.
What does number field sieve improve on quadrative sieve? I know we work with number fields and other details. However what exact complexity reason are we able to improve and why cannot we improve any further?
I know wiki has a few sentence explanation in comments. However it is difficult to pinpoint both why quadratic sieve was beaten and why no other alternative has been known for two decades.

In other words why does looking at abelian extensions of number fields help? 
Which extensions help and why?

REPLY [4 votes]: Each relation in the quadratic sieve has to factor integers of size near the square root of N. In contrast, each relation in the number field sieve has to factor two things, a number and an element of a number field. That number field can be chosen so that the two factorizations are much easier than the factorization in the quadratic sieve. The number field is chosen in terms of a polynomial of degree d, and as N increases in size the value of d is increased. The asymptotic complexity of NFS assumes the optimal value of d