Category:Definitions/Discriminants of Polynomials

This category contains definitions related to Discriminants of Polynomials.
Related results can be found in Category:Discriminants of Polynomials.

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:

$\ds \map \Delta f := \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {\alpha_i - \alpha_j}^2$