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Over the real numbers, a discriminant is equivalent to −1, 0, or 1. Over the rational numbers , a discriminant is equivalent to a unique square-free integer . By a theorem of Jacobi , a quadratic form over a field of characteristic different from 2 can be expressed, after a linear change of variables, in diagonal form as
An integral basis is given by {1, α, α(α + 1)/2} and the discriminant of K is −503. [5] [6] Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for higher-degree number fields. For example, there are two non-isomorphic cubic fields of discriminant 3969.
Its discriminant as quadratic form need not be +1 (in fact this happens only for the case K = Q). Define the inverse different or codifferent [ 3 ] [ 4 ] or Dedekind's complementary module [ 5 ] as the set I of x ∈ K such that tr( xy ) is an integer for all y in O K , then I is a fractional ideal of K containing O K .
The following table shows some orders of small discriminant of quadratic fields. The maximal order of an algebraic number field is its ring of integers, and the discriminant of the maximal order is the discriminant of the field. The discriminant of a non-maximal order is the product of the discriminant of the corresponding maximal order by the ...
Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or events.
The oldest problem in the theory of binary quadratic forms is the representation problem: describe the representations of a given number by a given quadratic form f. "Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of representations, or even just determine whether any ...
The next simplest example is the ring of Gaussian integers [], consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field Q ( i ) {\displaystyle \mathbb {Q} (i)} of Gaussian rationals , consisting of complex numbers whose real and imaginary parts are rational numbers.
τ(p r + 1) = τ(p)τ(p r) − p 11 τ(p r − 1) for p prime and r > 0. | τ ( p ) | ≤ 2 p 11/2 for all primes p . The first two properties were proved by Mordell (1917) and the third one, called the Ramanujan conjecture , was proved by Deligne in 1974 as a consequence of his proof of the Weil conjectures (specifically, he deduced it by ...