Basic Quadratic Forms
This book contains an introduction to arithmetics of quadratic forms and applications covering fundamentals of the theory of quadratic spaces and quadratic lattices over principal ideal domains. To read the book, only basic courses in abstract algebra are required; the book is essentially self-contained. As such, it can be used as a preparatory text devoted to deeper applications of quadratic forms. The author’s intention is also implied in the number of exercises and notices for further reading spread throughout the text. The part devoted to local-global theory starts with an introduction to valuation theory and p-adic numbers and ends with the Hasse-Minkowski theorem. It is followed by global integral theory, local classification of lattices and the local-global approach to global lattices. The final chapter has some applications of quadratic forms in cryptography and the description of the LLL-algorithm for finding short vectors in a lattice. The appendix, containing characterisations of 35 books on quadratic forms published between 1950 and 2007, is unusual but interesting. The book follows its aims in a straightforward manner omitting lengthy discussions. The goal of the author was to write a textbook appropriate for graduate courses or for independent study. This goal was certainly achieved.