Quadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modules are quadratic lattices (sometimes, simply lattices). They play an important role in number theory and topology.
An integral quadratic form has integer coefficients, such as x2 + xy + y2; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning Q(x,y) ∈ Z if x,y ∈ Λ.
This is the current use of the term; in the past it was sometimes used differently, as detailed below.
Read more about this topic: Quadratic Form
Famous quotes containing the words integral and/or forms:
“Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made mea book consubstantial with its author, concerned with my own self, an integral part of my life; not concerned with some third-hand, extraneous purpose, like all other books.”
—Michel de Montaigne (15331592)
“While waiting to get married, several forms of employment were acceptable. Teaching kindergarten was for those girls who stayed in school four years. The rest were secretaries, typists, file clerks, or receptionists in insurance firms or banks, preferably those owned or run by the family, but respectable enough if the boss was an upstanding Christian member of the community.”
—Barbara Howar (b. 1934)