Math 2.0

Field theory and polynomials Definition

Discriminant of a polynomial

Builds on Split polynomialFundamental theorem of symmetric polynomialsRepeated roots and the derivative

Definition

Let KK be a field and let f=anxn+⋯+a0∈K[x]f = a_n x^n + \dots + a_0 \in K[x] with n≥1n \ge 1 and an≠0a_n \neq 0. Let f=an(x−r1)⋯(x−rn)f = a_n (x - r_1) \cdots (x - r_n) in a splitting field of ff. The discriminant of ff is Disc⁡(f)=an2n−2∏1≤i<j≤n(ri−rj)2.\operatorname{Disc}(f) = a_n^{2n-2} \prod_{1 \le i < j \le n} (r_i - r_j)^2.

Remarks

The product is symmetric in the roots, so Disc⁡(f)∈K\operatorname{Disc}(f) \in K, by the fundamental theorem of symmetric polynomials. It is a polynomial with integer coefficients in a0,…,ana_0, \dots, a_n. So Disc⁡(f)=0\operatorname{Disc}(f) = 0 if and only if ff has a repeated root. For f=ax2+bx+cf = a x^2 + b x + c, Disc⁡(f)=b2−4ac\operatorname{Disc}(f) = b^2 - 4ac. For f=x3+px+qf = x^3 + p x + q, Disc⁡(f)=−4p3−27q2\operatorname{Disc}(f) = -4p^3 - 27q^2.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionPolynomials in several variables, definition, no evidence levelPolynomials in severa…definitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionFormal derivative, definition, no evidence levelFormal derivativedefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionSymmetric polynomial, definition, no evidence levelSymmetric polynomialdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionElementary symmetric polynomials, definition, no evidence levelElementary symmetric …definitionMultiplicity of a root, definition, no evidence levelMultiplicity of a rootdefinitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionFundamental theorem of symmetric polynomials, statement, no evidence levelFundamental theorem o…statementRepeated roots and the derivative, statement, no evidence levelRepeated roots and th…statementDiscriminant of a polynomial, definition, no evidence levelDiscriminant of a pol…definition
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Polynomials in several variables definition
  6. Degree of a polynomial definition
  7. Formal derivative definition
  8. Polynomial evaluation definition
  9. Symmetric polynomial definition
  10. Root of a polynomial definition
  11. Elementary symmetric polynomials definition
  12. Multiplicity of a root definition
  13. Split polynomial definition
  14. Fundamental theorem of symmetric polynomials statement
  15. Repeated roots and the derivative statement
  16. Discriminant of a polynomial definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Discriminant, Expression in terms of the roots
  2. Web https://en.wikipedia.org/wiki/Discriminant, Low degrees
About this page
Tag0CCPN09EMJ
URIm2://0CCPN09EMJ
Kinddefinition
WritingW1 stated
MSC12E05
DefinesDiscriminant of a polynomial
UsesSplit polynomial, Fundamental theorem of symmetric polynomials, Repeated roots and the derivative

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