Q:

Finding quadratic residues in a finite field by using a primitive element

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?

A:

$$\begin{array}{rcl}
\left(1+2x\right)^0 & = & 1 \\
\left(1+2x\right)^1 & = & 1+2x \\
\left(1+2x\right)^2 & = & x \\
\left(1+2x\right)^3 & = & 1+x \\
\left(1+2x\right)^4 & = & 2 \\
\left(1+2x\right)^5 & = & 2+x \\
\left(1+2x\right)^6 & = & 2x \\
\left(1+2x\right)^7 & = & 2x+2 \\
\end{array}$$
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 $1+2x$.

