The theory of quadratic forms has a long and glorious history: launched
in ancient Babylonia between 1900 and 1600 BC, taken up again by Brah-
magupta in the Seventh Century, and then—another thousand years later—
by the great genius Fermat, followed by a succession of extraordinary math-
ematicians, including Euler, Lagrange, and Gauss, who brought the subject
closer to its modern form. The work of Minkowski in the late Nineteenth
Century, coupled with the extension of his work by Hasse in the early Twen-
tieth Century, led to a great broadening and deepening of the theory that has
served as the foundation for an enormous amount of research that continues
Though the roots of the subject are in number theory of the purest
sort, the last third of the Twentieth Century brought with it new links of
quadratic forms to group theory, topology, and—most recently—to cryptog-
raphy and coding theory. So there are now many members of the mathe-
matical community who are not fundamentally number theorists but who
find themselves needing to learn about quadratic forms, especially over the
integers. There is thus a need for an accessible introductory book on qua-
dratic forms that can lead readers into the subject without demanding a
heavy background in algebraic number theory or previous exposure to a lot
of sophisticated algebraic machinery. My hope is that this is such a book.
One of the special attributes of number theory that distinguishes it from
most other areas of mathematics is that soon after a subject is introduced
and objects are defined, questions arise that can be understood even by a
newcomer to the subject, although the answers may have eluded the experts
for centuries. Even though this is an introductory book, it contains a sub-
stantial amount of material that has not yet appeared in book form, and
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