THIS is the first volume of Prof. Hancock's work and is intended as an introduction to the second volume. Its size shows that he is engaged in no light task. His object is to help in making the theory of algebraic numbers more accessible, more attractive, and less difficult. He obviously has in mind the English-speaking student, or at any rate those for whom German is a source of difficulty. For these, he has undoubtedly made more accessible and more easily readable the difficult theories of Dedekind and Kronecker, as well as the more elementary topics such as algebraic integers, quadratic and cubic fields, an introductory account of ideals, Hilbert's norm residue symbol, the law of quadratic reciprocity, and applications to Fermat's last theorem. The beginner will find much to interest him, as many of the chapters are selfcontained and will be useful in supplementing a course of lectures. Foundations of the Theory of Algebraic Numbers. Prof. Harris Hancock. Vol. 1: Introduction to the General Theory. (Published with the aid of the Charles Phelps Taft Memorial Fund, University of Cincinnati.) Pp. xxvii + 602. (New York: The Macmillan Co., 1931.) 8 dollars.