Find an element that generates the Ideal

Question

Given the ideal: $A$:={$\sum_{i∈I}T^i|$ $I $ finite,$|I|$ even} ⊂ $\mathbb{F}_2[X]$ find an element that generates it.

Now I know that it has to be $A=a$ $\mathbb{F}_2[X]$, where $a$ is the element that I'm looking for and $a≠0$.

Has it any sense to let $a=1$? but then $A=\mathbb{F}_2[X]$. I'm not really sure what I should look at first

Answer 1

Hint: Note that for any $p(T)=\sum_{i\in I}T^i\in A$, it holds $p(1)=|I|=0$. This gives $$ (T-1)=(T+1)\ | \ p(T) $$ for any $p\in A$.