Primitive elements of finite field with characteristic 2

Question

How would you find primitive elements of the finite field $\mathbb{F}_{2^c}$ for arbitrary integer $c\geq 1$? Is there some general way of doing this? I in fact only want to find one for each $c$ if there happens to be one that is simple to compute.

Answer 1

If you don't require a specific element or polynomial to generate the field you could look at the $(2^{c}-1)^{st}$ cyclotomic polynomial over $\Bbb{F}_2$. Any of its roots should be a primitive root of 1 and generate the multiplicative group of the field with $2^c$ elements.