This book, together with its sequel, “Further Algebra and Applications” (to appear), is a new and revised version of the author's famous text “Algebra (Vols. 2 and 3)”, which is now out of print. Besides basics on groups, rings, modules and fields, the book contains classical Galois theory of equations, an introduction to quadratic forms and ordered fields, valuation theory, and classical commutative ring theory (including primary decomposition and Hilbert Nullstellensatz). The final chapter deals with infinite field extensions. The book contains numerous exercises, and moreover, motivation and many illuminating comments on the subject. There is no doubt that the book will take the position of its predecessor in being one of the most outstanding introductory algebra textbooks.