<s>[INST] Find out the primitive polynomial GF(3) 1.) $x^2 + 2x$
2.) $x^2 + 1$
3.) $x^2 + 2$
4.) $x^2 + 2x$
5.) $x^2 + 2x + 1$
6.) $x^2 + 2x + 2$
7.) $x^2 $
8.) $x^2 + x + 2$
9.) $x^2 + x + 1$
Can any one help me in listing out primitive polynomials and tell me why is it a primitive polynomial please.
 [/INST] $$x^2 + 1,
x^2 + 2x + 2,
x^2 + x + 2,$$
are primitive polynomials over $gf(3)$.
Because they are irreducible and monic polynomials
Monic means coefficient of highest exponent of $x$ is $1$.
And irreducible means can not be further factorise into small polynomials.
Let's take first case $x^ 2 + 2x$ is further factorise as $x(x + 2)$.
Third case $x^2 + 2$ is monic but it is factorise as $(x-1)(x+1)$.
Because in $gf(3), 2 = -1$ and hence $x^2+2 = x^2 -1 = (x-1)(x+1)$.
And you can also check it by taking $x = 1$.  
$1^2 + 2 = 3$ which equal to $0$ ( because $3 \pmod3 = 0$) so $x-1$ is factor of $x^2 + 2$  in $gf(3)$.
Now check other polynomials by yourself and get answer.
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