## More on fields and polynomial

Have just learned that:

$F_{2}=Z_{2}=\left\{ \bar{0},\bar{1}\right\}$

$F_{4}=\left\{\left(a,b: a,b \epsilon F_{2}\right) \right\}$ means that the elements are

$\left(\bar{0},\bar{0} \right)$, $\left(\bar{0},\bar{1} \right)$, $\left(\bar{1},\bar{0} \right)$ and $\left(\bar{1},\bar{1} \right)$

$F_{8}=F_{2^{3}}\left\{\left(a, b, c: a, b, c \epsilon F_{2} \right) \right\}$

$F_{2^{n}}$ means that we’ll have $n$ tuple, thus we’ll have a polynomial with $n$ degree

(degree is the highest exponent of the polynomial).

Example:

$\left(\bar{1}, \bar{0}, \bar{1}, \bar{1} \right)$ means $x^{3}+x+1$

Important notes:

$F_{p}=Z_{p}$ is only if $p$ is prime. $F_{8}\neq Z_{8}$