Discriminant of a polynomial

Definition

Suppose $K$ is a field and $f(x)\in K[x]$ is a nonconstant polynomial. The discriminant of $f$ is defined in the following equivalent ways:

1. It is the resultant polynomial of $f(x)$ and its formal derivative $f'(x)$.
2. Let $L$ be a splitting field for $f$ over $K$. Over this, write:

$f(x)=a^{n}\prod _{i=1}^{n}(x-\alpha _{i})$.

Then, the discriminant of $f$ is defined as:

$a^{n}\prod _{1\leq i<j\leq n}(\alpha _{i}-\alpha _{j})^{2}$.

The discriminant of a linear polynomial $ax+b$ is defined to be $a$, because the product in this case is empty.

The discriminant of a polynomial is nonzero if and only if the polynomial is a separable polynomial.