Definition:Discriminant of Polynomial
Definition
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Let $k$ be a field.
Let $\map f X \in k \sqbrk X$ be a polynomial of degree $n$.
Let $\overline k$ be an algebraic closure of $k$.
Let the roots of $f$ in $\overline k$ be $\alpha_1, \alpha_2, \ldots, \alpha_n$.
Then the discriminant $\map \Delta f$ of $f$ is defined as:
- $\ds \map \Delta f := \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {\alpha_i - \alpha_j}^2$
Quadratic Equation
The concept is usually encountered in the context of a quadratic equation $a x^2 + b x + c = 0$:
The expression $b^2 - 4 a c$ is called the discriminant of the equation.
Cubic Equation
In the context of a cubic equation $a x^3 + b x^2 + c x + d = 0$:
Let:
- $Q = \dfrac {3 a c - b^2} {9 a^2}$
- $R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$
The discriminant of the cubic equation is given by:
- $D := Q^3 + R^2$
Examples
Quadratic
Let $\alpha_1$ and $\alpha_2$ be the roots of a quadratic equation $f$.
The discriminant of $f$ is:
- $\paren {\alpha_1 - \alpha_2}^2$
Cubic
Let $\alpha_1$, $\alpha_2$ and $\alpha_3$ be the roots of a cubic equation $f$.
The discriminant of $f$ is:
- $\paren {\alpha_1 - \alpha_2}^2 \paren {\alpha_2 - \alpha_3}^2 \paren {\alpha_3 - \alpha_1}^2$
Also see
- Results about discriminants of polynomials can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): discriminant (of a polynomial equation)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): discriminant (of a polynomial equation)