Vipul: Created page with '==Definition== Suppose $K$ is a field and $f(x) \in K[x]$ is a nonconstant polynomial. The '''discriminant''' of $f$ is defined in the follo...'

2009-05-15T01:59:20Z

Created page with '==Definition== Suppose <math>K</math> is a field and <math>f(x) \in K[x]</math> is a nonconstant polynomial. The '''discriminant''' of <math>f</math> is defined in the follo...'

New page

==Definition==

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

# It is the [[defining ingredient::resultant polynomial]] of <math>f(x)</math> and its [[defining ingredient::formal derivative of a polynomial|formal derivative]] <math>f'(x)</math>.
# Let <math>L</math> be a splitting field for <math>f</math> over <math>K</math>. Over this, write:

<math>f(x) = a^n \prod_{i=1}^n (x - \alpha_i)</math>.

Then, the discriminant of <math>f</math> is defined as:

<math>a^n \prod_{1 \le i < j \le n} (\alpha_i - \alpha_j)^2</math>.

The discriminant of a linear polynomial <math>ax + b</math> is defined to be <math>a</math>, 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]].

Discriminant of a polynomial - Revision history

Vipul: Created page with '==Definition== Suppose K is a field and f(x) \in K[x] is a nonconstant polynomial. The '''discriminant''' of f is defined in the follo...'

Vipul: Created page with '==Definition== Suppose $K$ is a field and $f(x) \in K[x]$ is a nonconstant polynomial. The '''discriminant''' of $f$ is defined in the follo...'