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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 ...
Explicitly it is the modular discriminant Δ ( z , q ) , {\displaystyle \Delta (z,q),} which represents (up to a normalizing constant ) the discriminant of the cubic on the right side of the Weierstrass equation of an elliptic curve ; and the 24-th power of the Dedekind eta function .
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]
Mathematical chemistry [1] is the area of research engaged in novel applications of mathematics to chemistry; it concerns itself principally with the mathematical modeling of chemical phenomena. [2] Mathematical chemistry has also sometimes been called computer chemistry, but should not be confused with computational chemistry.
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G.
By definition, the different ideal δ K is the inverse fractional ideal I −1: it is an ideal of O K. The ideal norm of δ K is equal to the ideal of Z generated by the field discriminant D K of K. The different of an element α of K with minimal polynomial f is defined to be δ(α) = f′(α) if α generates the field K (and zero otherwise ...
The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871. [16] At this point, he already knew the relationship between the discriminant and ramification. [17] Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857. [18]
The discriminant of a polynomial equation, especially the quadratic equation: [7] [8] =. The area of a triangle = . The symmetric difference of two sets. A macroscopic change in the value of a variable in mathematics or science.