Editing Discriminant (section)

===Degree 3===
{{seealso|Cubic equation#Discriminant}}
[[File:Discriminant of cubic polynomials..png|thumb|The zero set of discriminant of the cubic {{math|''x''<sup>3</sup> + ''bx''<sup>2</sup> + ''cx'' + ''d''}}, i.e. points satisfying {{math|1=''b''<sup>2</sup>''c''<sup>2</sup> − 4''c''<sup>3</sup> − 4''b''<sup>3</sup>''d'' − 27''d''<sup>2</sup> + 18''bcd'' = 0}}.]]
The cubic polynomial <math>ax^3+bx^2+cx+d \,</math> has discriminant
:<math>b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd\,.</math><ref>{{cite web| url=https://brilliant.org/wiki/cubic-discriminant/| title=Cubic Discriminant {{!}} Brilliant Math & Science Wiki| access-date=2023-03-21}}</ref><ref>{{cite web| url=https://www.johndcook.com/blog/2019/07/14/discriminant-of-a-cubic/| title=Discriminant of a cubic equation| date=14 July 2019| access-date=2023-03-21}}</ref>

In the special case of a [[Depressed cubic#Depressed cubic|depressed cubic]] polynomial <math>x^3+px+q</math>, the discriminant simplifies to
:<math> -4p^3-27q^2\,.</math>

The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers, and negative if there is one real root and two complex conjugate roots.<ref>{{cite book
|title=Integers, polynomials, and rings
|first1=Ronald S.
|last1=Irving
|publisher=Springer-Verlag New York, Inc.
|year=2004
|isbn=0-387-40397-3
|url=https://books.google.com/books?id=B4k6ltaxm5YC&pg=PA154
|at=ch. 10 ex. 10.14.4 & 10.17.4, pp. 154–156}}</ref>

The square root of a quantity strongly related to the discriminant appears in the [[Cubic equation#General cubic formula|formulas for the roots of a cubic polynomial]]. Specifically, this quantity can be {{math|−3}} times the discriminant, or its product with the square of a rational number; for example, the square of {{math|1/18}} in the case of [[Cardano formula]].

If the polynomial is irreducible and its coefficients are rational numbers (or belong to a [[number field]]), then the discriminant is a square of a rational number (or a number from the number field) if and only if the [[Galois group]] of the cubic equation is the [[cyclic group]] of [[order (group theory)|order]] three.