Discriminant

A quantity computed from a polynomial’s roots or a number field’s integral basis that is unchanged by basis choice and encodes arithmetic or geometric structure.

Vandermonde determinant: A determinant whose rows or columns contain successive powers of variables; its value is a product of pairwise differences. Squaring this determinant of the roots gives the polynomial discriminant.
Number field: A finite extension of the rational numbers. Its ring of integers and trace pairing define the number-field discriminant.
Quadratic field: A number field of degree two over the rational numbers. Its fundamental discriminant determines its ring of integers and distinguishes its arithmetic behavior.
Polynomial derivative: The polynomial obtained by formally differentiating a polynomial term by term. It enters the resultant formula for the discriminant but is not itself an invariant of the roots.
Trace pairing: A bilinear form on a field extension defined by taking the field trace of products. The number-field discriminant is the determinant of this pairing on an integral basis.
Ring of integers: The integral closure of the integers inside a number field. The discriminant belongs to this ring’s arithmetic structure, not merely to an arbitrary field basis.
Dedekind zeta function: A function encoding the ideal-counting arithmetic of a number field. The field discriminant appears in its functional equation and analytic size estimates.
Norm: A scalar associated with a field element, given by the determinant of its multiplication map. The norm concerns one element, whereas the discriminant measures a whole basis through pairwise trace products.
Resultant: A polynomial expression in two polynomials’ coefficients that vanishes exactly when they share a root. The polynomial discriminant is expressible through the resultant of a polynomial and its derivative.
Polynomial: An algebraic expression built from a variable, coefficients, addition, and multiplication. A polynomial’s roots and coefficients determine its polynomial discriminant.

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