Vipul: /* Facts */

2009-05-15T01:52:39Z

Facts

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	The formal derivative also satisfies the following rule for multiplication, called the Leibniz rule:		The formal derivative also satisfies the following rule for multiplication, called the Leibniz rule:

	<math>(fg)'(x) = f'(x)g(x) + f(x)g'(x)</math>.		<math>\! (fg)'(x) = f'(x)g(x) + f(x)g'(x)</math>.

Vipul: Created page with '==Definition== Suppose $K$ is a field and $f(x) \in K[x]$ is a polynomial. The '''formal derivative''' of $f$ , denoted $f'$ , is d...'

2009-05-15T01:52:05Z

Created page with '==Definition== Suppose <math>K</math> is a field and <math>f(x) \in K[x]</math> is a polynomial. The '''formal derivative''' of <math>f</math>, denoted <math>f'</math>, is d...'

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==Definition==

Suppose <math>K</math> is a [[field]] and <math>f(x) \in K[x]</math> is a polynomial. The '''formal derivative''' of <math>f</math>, denoted <math>f'</math>, is defined as follows.

If <math>f(x) = \sum_{m=0}^n a_mx^m</math>, then <math>f'(x) = \sum_{m=1}^n ma_mx^{m-1}</math>.

Here, the <math>ma_m</math> is understood as <math>a_m</math> added to itself <math>m</math> times.

==Facts==

The formal derivative gives a <math>K</math>-linear map from <math>K[x]</math> to itself. When <math>K</math> has characteristic zero, the kernel of the map is <math>K</math>, whereas when <math>K</math> has characteristic <math>p</math>, the kernel of the map is <math>K[x^p]</math>.

The formal derivative also satisfies the following rule for multiplication, called the Leibniz rule:

<math>(fg)'(x) = f'(x)g(x) + f(x)g'(x)</math>.

Formal derivative of a polynomial - Revision history

Vipul: /* Facts */

Vipul: Created page with '==Definition== Suppose K is a field and f(x) \in K[x] is a polynomial. The '''formal derivative''' of f, denoted f', is d...'

Vipul: Created page with '==Definition== Suppose $K$ is a field and $f(x) \in K[x]$ is a polynomial. The '''formal derivative''' of $f$ , denoted $f'$ , is d...'