Derivative of a polynomial in finite fields

Question

In Introduction to finite fields and their applications by R.Lidl, the definition of the derivative for a polynomial such that ($a_i\in GF(q)$) $$f_{(x)}=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$

is $$f_{(x)}=na_n*x^{n-1}+\cdots +a_1 $$

It is not clear what the term $na_n$ actually means. We already know that $a_n$ is a member of the field. However, it is not clear what $n$ means. It can be the polynomial representation or the power representation.

What is the correct definition?

j4GGy · Accepted Answer · 2015-05-18T09:48:12.420

2

$n a_n$ usually refers to the canonical group action of $\mathbb{Z}$ on the field you're considering, i.e. $$n a_n = \underbrace{a_n + \ldots + a_n}_{n \text{ times}}$$

EDIT: If you insist on using a polynomial representation, maybe this site can help.

edited May 18 '15 at 09:48

answered May 18 '15 at 09:26

j4GGy

3,721

Thank you. However here, 2 is converted to the filed member and an FF multiplication is used. – shampoo May 18 '15 at 09:33
1

It's the same thing because $2 = 1+1$. – fkraiem May 18 '15 at 09:44

Derivative of a polynomial in finite fields

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