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  2. Discriminant - Wikipedia

    en.wikipedia.org/wiki/Discriminant

    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

  3. Discriminant of an algebraic number field - Wikipedia

    en.wikipedia.org/wiki/Discriminant_of_an...

    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.

  4. Different ideal - Wikipedia

    en.wikipedia.org/wiki/Different_ideal

    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 .

  5. Quadratic field - Wikipedia

    en.wikipedia.org/wiki/Quadratic_field

    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 ...

  6. Second partial derivative test - Wikipedia

    en.wikipedia.org/wiki/Second_partial_derivative_test

    In cases 1 and 2, the requirement that f xx f yy − f xy 2 is positive at (x, y) implies that f xx and f yy have the same sign there. Therefore, the second condition, that f xx be greater (or less) than zero, could equivalently be that f yy or tr( H ) = f xx + f yy be greater (or less) than zero at that point.

  7. Binary quadratic form - Wikipedia

    en.wikipedia.org/wiki/Binary_quadratic_form

    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 ...

  8. PayPal fixes outage that affected thousands worldwide - AOL

    www.aol.com/news/paypal-outage-affects-thousands...

    The company experienced a system issue that affected multiple products including account withdrawals, peer-to-peer payment service Venmo, online checkout and crypto. PayPal said the issue, which ...

  9. Iris flower data set - Wikipedia

    en.wikipedia.org/wiki/Iris_flower_data_set

    Scatterplot of the data set. The Iris flower data set or Fisher's Iris data set is a multivariate data set used and made famous by the British statistician and biologist Ronald Fisher in his 1936 paper The use of multiple measurements in taxonomic problems as an example of linear discriminant analysis. [1]