In this aspect, they are probably unsurpassed as excellent sources for serious courses in a modern doctoral program: Lang intended them for specifically that purpose, and this is certainly the case for Algebraic Number Theory. In contrast with, e.g., Borevich-Shafarevich, Cassels-Fröhlich, and Weil, Lang occupies a very different place. What most distinguishes the many books by Serge Lang is their specific focus on teaching the indicated subject to the prepared student. Each and every one of these books is in its own way fabulous and vital for the number theorist to learn his craft, and it is in this collection that we should fit the book under review. Algebraic Number Theory, by Cassels and Fröhlich. the aforementioned book by Weil, in which idèles and adèles are featured with gusto the pedagogical masterpiece, Number Theory, by Borevich and Shafarevich and of course the Brighton Conference Proceedings, a.k.a. Many of these are now considered standards, e.g. As the 20th century marched on and research on this and allied parts of number theory intensified and expanded, other books appeared. In any case, the game was afoot, and many authors began to contribute their own versions of the subject, including Hermann Weyl, Emil Artin, and Helmut Hasse. These included what André Weil characterized in his Basic Number Theory as the premier source for this material given in classical language, namely, Vorlesungen über die Theorie der algebraischen Zahlkörper, or Lectures on the Theory of Algebraic Number Fields, by Hilbert’s student Erich Hecke. So the stage was set: in proper order algebraic number theory could be taught as a well-defined university course, and texts began to emerge.
For another, Hilbert gave the definitive treatment of reciprocity laws, going well beyond the quadratic one (Gauss-Euler), covering all degrees and all algebraic base fields.
For one thing, he gave a very far-reaching extension of Kronecker’s work leading to modern class field theory. roots of polynomials with integer coefficients. Of course, Hilbert did a great deal more than collect and present the known number theoretic material surrounding algebraic numbers, i.e. However, it is undeniable that by the end of that century, this discipline was ready for a proper definition, and it was Hilbert’s famous Zahlbericht that did the job.
Indeed we encounter a host of players throughout the later 19th century, including Eisenstein, Sophie Germain, Dedekind, Dirichlet, Kummer, Kronecker, and so on: I am sure I am missing some obvious others. Gauss, with his famous Disquitiones Arithmeticae, is likely the one to be credited with giving the greatest impetus to what eventually became this algebraic theory of numbers as a Ding in Sich certainly, the subject, while still not autonomously defined, took off spectacularly in his wake. I believe that a proof can be glued together from results Euler had at his disposal, to be sure, but the first complete proof is credited to Gauss - who then gave six more. Its prehistory sports none other than Fermat and Euler: just think of Fermat’s Last Theorem and the Law of Quadratic Reciprocity, which Euler knew but never explicitly proved. Algebraic number theory has an unparalleled pedigree.
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