Quadratic forms are homogeneous functions of degree two in n variables. Specifically, they are functions of the form Σaijxixj.
They can be put in correspondence with symmetric n-by-n matrices and, when the characteristic of the field is not equal to two, with symmetric bilinear forms on vector spaces defined over the field. While these functions appear quite elementary, they have deep connections throughout mathematics, to areas such as the study of Clifford Algebras, K-Theory Galois Cohomology, and the Chow groups that are found in Algebraic Geometry. For anyone wishing to be introduced to the study of quadratic forms, I highly recommend T.Y. Lam's Introduction to Quadratic Forms over a Field
This book is a heavily updated version of the author's The Algebraic Theory of Quadratic Forms, originally published in 1972, and long one of the standard books in the area. The book starts at a level which most graduate students could follow, and dedicates a chapter to the foundations of quadratic forms and quadratic spaces before delving deeper into Witt Rings, Quaternion Algebras, and Clifford Algebras. Throughout these early chapters, the book works primarily over an arbitrary field of characteristic not equal to two, although the author supplies many detailed examples of what happens over, say, the real numbers. Later chapters present expositions of the study of quadratic forms over local fields and over global fields, as well as looking at what happens over algebraic extensions of fields, over function fields, and various other hypotheses. The final chapters are dedicated to "Special Topics" in quadratic forms as well as in invariants, such as Kaplansky Radicals, low-dimensional quadratic forms, and Pythagoras numbers.
The level of exposition in this book is quite high. I found Lam's explanations to be very clear — even when it felt like I had to duck to get out of the way of the large number of technical definitions he was throwing my way, I was able to successfully follow all that was going on. He provided a large number of examples and exercises which aided the exposition, and it was on the whole very chatty. It even included a level of wit (no pun intended) and humor that one rarely finds in a technical monograph. In particular, this was a book that I not only learned quite a bit from, but was also a pleasure to read.
Darren Glass is Assistant Professor of Mathematics at Columbia University.