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.
$$x^2 + 1, x^2 + 2x + 2, x^2 + x + 2,$$ 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.