# How to list all elements of finite field (e.g., GF(2^5)) with trace 1.

Note that trace is defined over finite field GF(2^n) as Tr(x)=\sum_{ i=1}^{k} X^{2^{n-1}}.

How to list all elements of finite field (e.g., GF(2^5)) with trace 1.

Note that trace is defined over finite field GF(2^n) as Tr(x)=\sum_{ i=1}^{k} X^{2^{n-1}}.

Please start posting anonymously - your entry will be published after you log in or create a new account.

Asked: ** 2017-08-14 14:26:19 +0200 **

Seen: **403 times**

Last updated: **Aug 14 '17**

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.

What did you try? Do you know how to list all elements of GF(2^5) in SageMath?

The lines

initialize the field $K$. Since the generator $a$ has minimal polynomial

its trace, resp. norm are unsurprisingly (coefficients in degrees four and zero)

The list of the elements of

normone can be computed by list comprehension, just take a glance at:and the mathematical version of it: $${ \ x \ : \ x\in K\text{ with norm}(x) = 1\ }\ .$$

Which is the command line for the list of elements of trace one in $K$?

How many elements of trace one are in $K$?

How many elements of norm one are in $K$?