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If the discriminant is positive, the number of non-real roots is a multiple of 4. That is, there is a nonnegative integer k ≤ n/4 such that there are 2k pairs of complex conjugate roots and n − 4k real roots. If the discriminant is negative, the number of non-real roots is not a multiple of 4.
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.
Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or events.
For example, 15 is a composite number because 15 = 3 · 5, but 7 is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4).
In algebraic number theory, the different ideal (sometimes simply the different) is defined to measure the (possible) lack of duality in the ring of integers of an algebraic number field K, with respect to the field trace. It then encodes the ramification data for prime ideals of the ring of integers. It was introduced by Richard Dedekind in 1882.
The discriminant of a polynomial is a quantity that depends on the coefficients and determines various properties of the roots. Discriminant may also refer to its various generalizations: Mathematics
An integer D is called a fundamental discriminant if it is the discriminant of a quadratic number field. This is equivalent to D ≠ 1 and either a) D is squarefree and D ≡ 1 (mod 4) or b) D ≡ 0 (mod 4), D /4 is squarefree, and D /4 ≡ 2 or 3 (mod 4).
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 ...