# 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}}.

Asked: **
2017-08-14 07:26:19 -0500
**

Seen: **322 times**

Last updated: **Aug 14 '17**

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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 $K$?