Calculating the index of an aditive subgroup.

Question

Let $\mathbb{F}_{q}$ a finite field with characteristic $2$, and define over it the set $N=\{x^2+x:x\in\mathbb{F}_q\}$ show that $[\mathbb{F}_q:N]=2$.

I've already proof that $N$ is an aditive normal subgroup of $\mathbb{F}_q$ but, I don't have any idea of how to verify the value of the index using cosets, somebody can give me a hint?

Answer 1

Hint:

1. Show that $f:x\mapsto x+x^2$ is a homomorphism of additive groups from $\Bbb{F}_q$ to itself.
2. Determine the subgroup $\operatorname{ker} f$.
3. The first isomorphism theorem.