Quadratic Form - Integral Quadratic Forms

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:

    An island always pleases my imagination, even the smallest, as a small continent and integral portion of the globe. I have a fancy for building my hut on one. Even a bare, grassy isle, which I can see entirely over at a glance, has some undefined and mysterious charm for me.
    Henry David Thoreau (1817–1862)

    That food has always been, and will continue to be, the basis for one of our greater snobbisms does not explain the fact that the attitude toward the food choice of others is becoming more and more heatedly exclusive until it may well turn into one of those forms of bigotry against which gallant little committees are constantly planning campaigns in the cause of justice and decency.
    Cornelia Otis Skinner (1901–1979)