# Basis and dimension of Ring as a vector space over finite field

I tried to solve following exercise:

My knowledge in algebra is a bit rusty, so I would appreciate it if you could help me remembering how to solve c). I realized that $$|R| = 2^{128}$$ but I don't know how to proceed.

• Wait, $\Bbb F=\Bbb F_{256}$, isn't it? Anyway, to prove $R$ is a vector space over $\Bbb F$, you basically need to verify that multiplication with elements of $\Bbb F$ is well defined and well behaved with the addition. Jun 10, 2022 at 19:16
• Well, write up some elements of $R$ first, then try to write up a generic element. The dimension will be the number of $\Bbb F$-coefficients in a properly written generic element. To create a basis just put $1$ for a given coefficient and $0$ for the others. (Doing it with a general $\Bbb F$ might actually ease it up.) Jun 10, 2022 at 19:22
• Ohhhhh. I remember. Well. $dim_\mathbb{F} (R) = 16$ and basis consisting of monomials of degree $\leq 3$ if I calculated right. Jun 10, 2022 at 19:44