As I see it， the graduate course in algebra must primarily prepare studentsto handle the algebra which they will meet in all of mathematics： topology，partial differential equations， differential geometry， algebraic geometry， analysis，and representation theory， not to speak of algebra itself and algebraic numbertheory with all its ramifications. Hence I have inserted throughout references topapers and books which have appeared during the last decades， to indicate someof the directions in which the algebraic foundations provided by this book areused; I have accompanied these references with some motivating comments， toexplain how the topics of the present book fit into the mathematics that is tocome subsequently in various fields; and I have also mentioned some unsolvedproblems of mathematics in algebra and number theory. The abc conjecture isperhaps the most spectacular of these.