Binary code over $F_8$

Question

Let $f(x) = x^3 + x^2 + 1$ be irreducible polynomial factor of $x^7 - 1$. I want to prove the following

1. $f$ as exactly three roots in $\mathbb{F}_8^*$
2. let $a$ be some root of $f$, then $\forall g \in \mathbb{F}_2[x] : g(a) = 0$ implies $f \mid g$
3. The code generated by $f$ is $[7,4,3]_2$ code.

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So far, I have the following partial thoughts:

2. since $f$ is irreducible, it's also must be the minimum polynomial of $a$, otherwise it wasn't irreducible. Thus, it must be $f \mid g$

But regarding (1), (3) i have no idea, so any help will be appreciate!

Answer 1

A hint for a different route to part 3:

1. Show that the multiplicative order of $a$ is seven. In other words $a^n\neq1$ for all $n$, $0<n<7$.
2. Show that if $f$ is a factor of a binomial $g(x)=x^i+x^j$, $i>j$ then $g$ has degree at least $7$.