Construction of finite fields

Question

This is the correct question

This is my answer for 1.1

This is my answer for 1.2

I have renewed this question correctly and I have done 1.1. But in 1.2，I don’t know whether it is correct to fill this muliplication table. Many thanks!

Answer 1

Let $p=x^2+x+2$ (I assume there is a typo on your sheet). Let $X$ be the image of $x$ in $\mathbb F_3[x]/(p).$

1.) Show that $p$ has no roots (this suffices since p has degree 2). Since $p$ is irreducible it follows that $(p)$ is a maximal ideal (use that $F[x]$ is a PID for and field $F$).

2.) This is only some calculation. You can take some primitive element (in this case you can take $X$, see later), then just take powers of it and mod out $p$. So $X^2=1+2X$, $X^3=2+2X$ etc.

3.) Here you can factor the cyclotomic polynomial in $\mathbb F_3[x]$ and get $x^4+1=(2 + x + x^2) (2 + 2 x + x^2)$. So the roots of these two polynomials are the primitive elements of your fieldextension. From this you know that our $X$ is a primitve element. The generators of $C_8$ are $1,3,5,7$, so you know the primitve elements are $X^1,X^3,X^5,X^7$.