I tried to solve following exercise: enter image description here

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.

Thanks in advance!

  • 2
    $\begingroup$ 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. $\endgroup$
    – Berci
    Jun 10, 2022 at 19:16
  • $\begingroup$ I agree. That's what I realized so far, too. But how does it help me in finding a basis? Sry, I'm having a massive brain fart rn $\endgroup$
    – Anton2107
    Jun 10, 2022 at 19:18
  • $\begingroup$ 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.) $\endgroup$
    – Berci
    Jun 10, 2022 at 19:22
  • 1
    $\begingroup$ Ohhhhh. I remember. Well. $dim_\mathbb{F} (R) = 16$ and basis consisting of monomials of degree $\leq 3$ if I calculated right. $\endgroup$
    – Anton2107
    Jun 10, 2022 at 19:44
  • 1
    $\begingroup$ I'm stuck at question (a) $\endgroup$
    – reuns
    Jun 10, 2022 at 20:31


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