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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 ...
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 ...
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.
The case of the 105th cyclotomic polynomial is interesting because 105 is the least positive integer that is the product of three distinct odd prime numbers (3×5×7) and this polynomial is the first one that has a coefficient other than 1, 0, or −1: [3]
The discriminant of a polynomial is a function of its coefficients that is zero if and only if the polynomial has a multiple root, or, if it is divisible by the square of a non-constant polynomial. In other words, the discriminant is nonzero if and only if the polynomial is square-free.
In the case of a non-cyclic cubic field K this index formula can be combined with the conductor formula D = f 2 d to obtain a decomposition of the polynomial discriminant Δ = i(θ) 2 f 2 d into the square of the product i(θ)f and the discriminant d of the quadratic field k associated with the cubic field K, where d is squarefree up to a ...
Macaulay's resultant is a polynomial in the coefficients of these n homogeneous polynomials that vanishes if and only if the polynomials have a common non-zero solution in an algebraically closed field containing the coefficients, or, equivalently, if the n hyper surfaces defined by the polynomials have a common zero in the n –1 dimensional ...
This expression acts as a discriminant in the sense that it is zero if and only if there is a non-zero solution in six unknowns x i, y i, z i, (with superscript i = 0 or 1) of the following system of equations a 000 x 0 y 0 + a 010 x 0 y 1 + a 100 x 1 y 0 + a 110 x 1 y 1 = 0 a 001 x 0 y 0 + a 011 x 0 y 1 + a 101 x 1 y 0 + a 111 x 1 y 1 = 0 a ...