How do I add the elements of a finite field knowing the multiplicative structure?

Question

Let $F$ be a finite field then the multiplicative part $F^\times$ is a cyclic group generated by $f$.

What - when nonzero - is $f^i + f^j$ as a power of $f$? What is 1,2,3,4,.. in terms of $f$?

For example $\mathbb F_{2^2} = \mathbb F_2[X]/(X^2+X+1)$ is generated by $X$ and $X^1 + X^2 = X^3$.

The Wikipedia article on Zech logarithms was edited by yours truly, and may now be somewhat useful :-/ I'm fairly sure a more experienced author of wiki-articles has edited it since, but I take credit for getting rid of a lot of unrelated material, and adding an example. — Jyrki Lahtonen, Mar 24 '13 at 12:07

score 3 · Answer 1 · answered Mar 24 '13 at 04:24

3

I don't think you can tell, in general. A finite field may have many generators, and it is possible that for generators $f$ and $g$ we have $1+f=f^i$ and $1+g=g^j$ with $i\ne j$. Perhaps you could try to find an example of this phenomenon.

answered Mar 24 '13 at 04:24

Gerry Myerson

179,216

3

Just right. Another way of saying this is that an abstract cyclic group of order $q-1$ can be the multiplicative group of a finite field in many different ways (in fact $\varphi(q-1)$ ways!) – Lubin Mar 24 '13 at 04:31

How do I add the elements of a finite field knowing the multiplicative structure?

1 Answers1