Formal derivative ( $f^{'} (T)$ )

The formal derivative of

f (T) = \sum_{i} a_{i} T^{i}

is

f^{'} (T) = \sum_{i} i a_{i} T^{i - 1}

.

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- Formal derivative in positive characteristic Formal derivative
  - Polynomial with zero formal derivative in characteristic p Formal derivative in positive characteristic

Formal derivative in positive characteristic ( $f^{'} (T)$ )

In characteristic

p

, the coefficient

i

in a formal derivative vanishes exactly when

p

divides

i

.

Polynomial with zero formal derivative in characteristic p

Over a field of characteristic

p > 0

,

f^{'} (T) = 0 ⟺ f (T) = g (T^{p})

for some polynomial

g

. Indeed, zero derivative says every exponent with nonzero coefficient is divisible by

p

.

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Formal derivative by

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The formal derivative is a concept in algebra and polynomial theory that generalizes the notion of a derivative from calculus to polynomials. It allows us to differentiate polynomials and power series without considering their convergence or limit processes, operating instead purely within the realm of algebra.

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