Let $1+2x$ be a primitive element of the field $\mathbb F_9$ obtained via the irreducible polynomial
$$x^2 + 1$$
over the base field $\mathbb F_3$.
i) Make a list of the elements of $\mathbb F_9$ together with the primitive element $1+2x$ and all the powers of primitive element.
ii) Which powers are quadratic residues and which are quadratic non-residues? Why?
$$\begin{array}{rcl} \left(1+2x\right)^0 & = & 1 \\ \left(1+2x\right)^1 & = & 1+2x \\ \left(1+2x\right)^2 & = & x \\ \left(1+2x\right)^3 & = & 1+x \\ \left(1+2x\right)^4 & = & 2 \\ \left(1+2x\right)^5 & = & 2+x \\ \left(1+2x\right)^6 & = & 2x \\ \left(1+2x\right)^7 & = & 2x+2 \\ \end{array}$$ Since $1+2x$ is primitive, every element of $\mathbb{F}_9$ has the form $\left(1+2x\right)^\alpha$ for some $\alpha$. Thus $$\left(\left(1+2x\right)^\alpha\right)^2=\left(1+2x\right)^{2\alpha}$$ so the quadratic residues are $1,2,x,$ and $2x$ - the even powers of $1+2x$.