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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 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]
For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field () with x 3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place that does.
Let D be the discriminant of the field, n be the degree of K over , and = be the number of complex embeddings where is the number of real embeddings.Then every class in the ideal class group of K contains an integral ideal of norm not exceeding Minkowski's bound
All complex cubic fields with discriminant greater than −500 have class number one, except the fields with discriminants −283, −331 and −491 which have class number 2. The real root of the polynomial for −23 is the reciprocal of the plastic ratio (negated), while that for −31 is the reciprocal of the supergolden ratio .
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 ...
Its square is widely called the discriminant, though some sources call the Vandermonde polynomial itself the discriminant. The discriminant (the square of the Vandermonde polynomial: Δ = V n 2 {\displaystyle \Delta =V_{n}^{2}} ) does not depend on the order of terms, as ( − 1 ) 2 = 1 {\displaystyle (-1)^{2}=1} , and is thus an invariant of ...
Each genus is the union of a finite number of equivalence classes of the same discriminant, with the number of classes depending only on the discriminant. In the context of binary quadratic forms, genera can be defined either through congruence classes of numbers represented by forms or by genus characters defined on the set of forms.