Separable polynomial: Difference between revisions

Latest revision as of 23:29, 15 May 2009

Template:Univariate polynomial upto associates property

Definition

Let $K$ be a field and $f (x) \in K [x]$ be a nonzero polynomial. We say that $f$ is a separable polynomial if the following equivalent conditions are satisfied:

$f (x)$ and its formal derivative $f^{'} (x)$ are relatively prime in $K [x]$ .
$f (x)$ splits completely into distinct linear factors over its splitting field.
For any field $L$ containing $K$ , $f$ is a square-free polynomial over $L$ , i.e., no square of a polynomial divides $f$ .
The discriminant of $f$ is nonzero.

@@ Line 8: / Line 8: @@
 # <math>f(x)</math> splits completely into ''distinct'' linear factors over its splitting field.
 # For any field <math>L</math> containing <math>K</math>, <math>f</math> is a square-free polynomial over <math>L</math>, i.e., no square of a polynomial divides <math>f</math>.
+# The [[defining ingredient::discriminant of a polynomial|discriminant]] of <math>f</math> is nonzero.