# Galois fields

• Oct 24th 2010, 03:13 AM
maken83
Galois fields
Is there any computer program for finding the elements of GF(p^n), because manually this process is quit lengthy. If we want to find the elements of GF(41^2), it will take several hours. Pleas tell me computer program and command to find the elements of GF(p^n). Is there any list of the elements of GF(p^n) available in internet????
• Oct 24th 2010, 04:03 AM
tonio
Quote:

Originally Posted by maken83
Is there any computer program for finding the elements of GF(p^n), because manually this process is quit lengthy. If we want to find the elements of GF(41^2), it will take several hours. Pleas tell me computer program and command to find the elements of GF(p^n). Is there any list of the elements of GF(p^n) available in internet????

I wouldn't worry about the elements of $\displaystyle GF(p^n)$ but about irreducible polynomials of degree n over $\displaystyle GF(p)$ . Once you know

these the elements of $\displaystyle GF(p^n)$ are easy to write down.

For example, using Gauss' quadratic law we know that $\displaystyle x^2-3$ is irreducible over $\displaystyle GF(41)$, so the elements

in $\displaystyle GF(41^2)$ are all the elements of the form $\displaystyle a+b\omega$ , with $\displaystyle \omega^2=3\,,\,\,a,b\in GF(41)$...pretty easy!

Tonio