<s>[INST] Determining whether a base is a Generator in $Z^*_{11}$ as well as differentiating Primitive Root and/or Primitive Element? Given $Z^*_{11}$, specifically looking at the bases $2$ and $3$:
\begin{array} {|r|r|}\hline 2^1 \pmod {11}=2 & 3^1 \pmod {11}=3 \\ \hline 2^2 \pmod {11}=4 & 3^2 \pmod {11}=9 \\ \hline 2^3 \pmod {11}=8 & 3^3 \pmod {11}=5 \\ \hline 2^4 \pmod {11}=5 & 3^4 \pmod {11}=4 \\ \hline 2^5 \pmod {11}={10} & 3^5 \pmod {11}=1 \\ \hline 2^6 \pmod {11}=9 & 3^6 \pmod {11}=3 \\ \hline 2^7 \pmod {11}=7 & 3^7 \pmod {11}=9 \\ \hline 2^8 \pmod {11}=3 & 3^8 \pmod {11}=5 \\ \hline 2^9 \pmod {11}=6 & 3^9 \pmod {11}=4 \\ \hline 2^{10} \pmod {11}=1 & 3^{10} \pmod {11}=1 \\ \hline  \end{array}
$ord_{11}(2)=10$ which is equivalent to $φ(11)=10$, so $k$ is the smallest integer
in $2^k \equiv 1 \pmod {11}$.
$ord_{11}(3)=5$, where $5$ is the smallest $k$ 
in $3^k \equiv 1 \pmod {11}$.
I'm fairly certain that $2$ is a generator, however I don't think $3$ is a generator because it doesn't generate every power in $Z^*_{11}$? But I've seen some posts that conflict with this. If $2$ is a generator, is it also a primitive root? I've also seen the term primitive element and wonder if it is synonomous as well?
 [/INST] $x$ is a primitive root (synonymously, a primitive element) mod $n$ if and only if
$x$ is a generator of the multiplicative group of integers modulo $n$.
Modulo $11$, $2$ is a primitive root and $3$ is not.
Modulo $7$, on the other hand, $3$ is a primitive root and $2$ is not.
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