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I would like to learn to solve Diophantine equations and I think my next step would be quadratic fields or number fields. What are kind of methods there are to use those on solving equations? And what kind of Diophantine equation can be solved using quadratic fields? Are there just few methods one can list here or should I read some book?

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You should read a book on algebraic number theory, e.g. Stewart and Tall or Neukirch. – Alex B. Dec 3 '11 at 12:16
Two books that explicitly deal with the connection: Cox's Primes of the Form $x^2 + ny^2$ and Frölich-Taylor's Algebraic Number Theory. – Dylan Moreland Dec 3 '11 at 22:24

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