Heights and diophantine equations over number fields

In the previous chapters we have worked essentially with ‘classically’ integral solutions, that is over ℤ. However, since the times of Kummer (and even of Gauss) it has been recognized that diophantine equations are most advantageously dealt with by going out of ℚand using tools from Algebraic Number Theory; this also led to consider solutions in integers of number fields, and even in S -integers therein, i.e . those which have a denominator composed only of primes in the finite set S . In turn, new concepts have been created for studying these more general solutions. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.