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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. Minkowski's bound - Wikipedia

    en.wikipedia.org/wiki/Minkowski's_bound

    Since the number of integral ideals of given norm is finite, the finiteness of the class number is an immediate consequence, [1] and further, the ideal class group is generated by the prime ideals of norm at most M K. Minkowski's bound may be used to derive a lower bound for the discriminant of a field K given n, r 1 and r 2.

  4. Discriminant of an algebraic number field - Wikipedia

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

    The discriminant of K is 49 = 7 2. Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7. In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field.

  5. 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 .

  6. Cubic field - Wikipedia

    en.wikipedia.org/wiki/Cubic_field

    It has the smallest discriminant of all totally real cubic fields, namely 49. [4] The field obtained by adjoining to Q a root of x 3 + x 2 − 3x − 1 is an example of a totally real cubic field that is not cyclic. Its discriminant is 148, the smallest discriminant of a non-cyclic totally real cubic field. [5]

  7. 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 ...

  8. Resolvent (Galois theory) - Wikipedia

    en.wikipedia.org/wiki/Resolvent_(Galois_theory)

    The simplest examples of resolvents are X 2 − Δ {\displaystyle X^{2}-\Delta } where Δ {\displaystyle \Delta } is the discriminant , which is a resolvent for the alternating group . In the case of a cubic equation , this resolvent is sometimes called the quadratic resolvent ; its roots appear explicitly in the formulas for the roots of a ...

  9. Second partial derivative test - Wikipedia

    en.wikipedia.org/wiki/Second_partial_derivative_test

    Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at (,) since = <. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point.