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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 ...
which is exactly the definition of the discriminant of the minimal polynomial. Let K = Q(α) be the number field obtained by adjoining a root α of the polynomial x 3 − x 2 − 2x − 8. This is Richard Dedekind's original example of a number field whose ring of integers does not possess a power basis.
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. If r 1, r 2, r 3 are the three roots (not necessarily ...
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 ...
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.
By contrast, the discriminant () does not depend on any order, so that Galois theory implies that the discriminant is a polynomial function of the coefficients of (). The determinant formula is proved below in three ways.
Hurwitz polynomial; Polynomial transformation; Tschirnhaus transformation; Galois theory; Discriminant of a polynomial. Resultant; Elimination theory. Gröbner basis; Regular chain; Triangular decomposition; Sturm's theorem; Descartes' rule of signs; Carlitz–Wan conjecture; Polynomial decomposition, factorization under functional composition
The roots, stationary points, inflection point and concavity of a cubic polynomial x 3 − 6x 2 + 9x − 4 (solid black curve) and its first (dashed red) and second (dotted orange) derivatives. The critical points of a cubic function are its stationary points , that is the points where the slope of the function is zero. [ 2 ]