I need a projection from $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{n-1}}$. I was thinking in a projection of vector spaces, but i want to know if there is a "canonical" projection or something like that, because tha will be very helpful.
Maybe i should reformulate my question: i would like to find some kind of mapping from $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{n-1}}$ x $\mathbb{F}_{2}$
On
You must remember that $\mathbb F(p^{n-1})$ is not contained in $\mathbb F(p^n)$, except of course in the case $n=2$. So I wonder just what you meant by the word “projection”. Did you just mean a surjective vector-space morphism (as $\mathbb F(p)$-spaces)? That’s all you can hope for, and there’s certainly nothing canonical, except to try to insure that the elements of the intersection-field $\mathbb F(p)$ are left fixed.
Yes, there is a canonical projection, which forgets the last coordinate.
Note that there can be only such a projection that is a homomorphism of vector spaces and not that of fields, because every ring homomorphism $K\to R$ from a field is injective or constant $0$.