The abc-conjecture for Algebraic Numbers

The abc.conjecture for the ring of integers states that, for every . > 0 and every triple of relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) . rad(abc)1 + . with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors of m. In the present paper we propose an abc.conjecture for the field of all algebraic numbers. It is based on the definition of the radical (in Section 1) and of the height (in Section 2) of an algebraic number. From this abc.conjecture we deduce some versions of Fermat's last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat's equation in fields of small degrees over ..