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In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic ...
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.
From a modern perspective, the class group of a fundamental discriminant is isomorphic to the narrow class group of the quadratic field of discriminant . [6] For negative Δ {\displaystyle \Delta } , the narrow class group is the same as the ideal class group , but for positive Δ {\displaystyle \Delta } it may be twice as big.
Since the sign of the discriminant of a number field K is (−1) r 2, where r 2 is the number of conjugate pairs of complex embeddings of K into C, the discriminant of a cubic field will be positive precisely when the field is totally real, and negative if it is a complex cubic field.
Rogers Discriminant Function; to assist in identifying simulated profiles with a negative bias. Further identification of exaggeration and/or negative bias can be calculated used the NIM Predicted Profile [1] Additionally, one can also apply the use of the Negative Distortion Scale [2]
The discriminant Δ of the cubic is the square of = () (), where a is the leading coefficient of the cubic, and r 1, r 2 and r 3 are the three roots of the cubic. As Δ {\displaystyle {\sqrt {\Delta }}} changes of sign if two roots are exchanged, Δ {\displaystyle {\sqrt {\Delta }}} is fixed by the Galois group only if the Galois group is A 3 .
The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in K / (K ×) 2 (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only "positive, zero, or negative".
(Although the factor −16 is irrelevant to whether or not the curve is non-singular, this definition of the discriminant is useful in a more advanced study of elliptic curves.) [2] The real graph of a non-singular curve has two components if its discriminant is positive, and one component if it is negative. For example, in the graphs shown in ...