Alphabeta MathHow statement and proof provenance work
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The discriminant of a monic polynomial as the coefficient expression of
Definition
By The square of the Vandermonde polynomial is symmetric and Fundamental theorem of symmetric polynomials: unique expression as a polynomial in , there is a unique polynomial such that
For a monic polynomial
over a commutative ring, its discriminant is
Equivalently, in any algebra in which splits with roots , this coefficient expression evaluates to . The definition therefore depends only on the coefficients and not on a choice or ordering of roots. For a monic constant polynomial, .
Depends on
- The square of the Vandermonde polynomial is symmetric
- Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_1,\ldots,e_n$
- A symmetric polynomial in the roots of a monic polynomial is a polynomial in its coefficients and lies in the base ring
- Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree
Used by
- Frobenius in a small cyclotomic field Example
- x³-3x+1 has discriminant 81 and Galois group A₃ over ℚ Example
- Good polynomial reduction kills inertia Lemma
- The coefficient formula and discriminant of the quartic resolvent Proposition
- For a monic separable polynomial in characteristic not two, the Galois group lies in Aₙ exactly when the discriminant is a square Theorem
- The discriminant is ∏_i<j(αᵢ-αⱼ)² and vanishes exactly when a monic polynomial has a repeated root Theorem
Dependency tree · two levels
11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. S. Milne, Fields and Galois Theory, Proposition 4.35 through Example 4.37 (standard reference, not scraped)