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An integral quadratic form has integer coefficients, such as x 2 + xy + y 2; 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.
In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise it is a definite quadratic form. More explicitly, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0.
If one takes the other cyclotomic fields, they have Galois groups with extra -torsion, so contain at least three quadratic fields. In general a quadratic field of field discriminant can be obtained as a subfield of a cyclotomic field of -th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of ...
The Hasse–Minkowski theorem reduces the problem of classifying quadratic forms over a number field K up to equivalence to the set of analogous but much simpler questions over local fields. Basic invariants of a nonsingular quadratic form are its dimension , which is a positive integer, and its discriminant modulo the squares in K , which is ...
The Cartan–Dieudonné theorem is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two. The reflection through the origin (the map v ↦ −v) is an example of an element of O(n) that is not a product of fewer than n reflections.
Certain invariants of a quadratic form can be regarded as functions on Witt classes. Dimension mod 2 is a function on classes: the discriminant is also well-defined. The Hasse invariant of a quadratic form is again, a well-defined function on Witt classes with values in the Brauer group of the field of definition. [22]
A complex number is called a quadratic integer if it is a root of some monic polynomial (a polynomial whose leading coefficient is 1) of degree two whose coefficients are integers, i.e. quadratic integers are algebraic integers of degree two. Thus quadratic integers are those complex numbers that are solutions of equations of the form x 2 + bx ...
An indefinite quadratic form takes on both positive and negative values and is called an isotropic quadratic form. More generally, these definitions apply to any vector space over an ordered field . [ 1 ]