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 me—a 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 (1533–1592)
“Your body must become familiar with its death—in all its possible forms and degrees—as a self-evident, imminent, and emotionally neutral step on the way towards the goal you have found worthy of your life.”
—Dag Hammarskjöld (1905–1961)