How are the fields $\mathbb{F}_k$(where $k$ is an integer) be generated?

Question

What are elements like in the fields $\mathbb{F}_k$? Does $\mathbb{F}_k$ contain only $k$ elements? When $k$ is a composite integer, what will be different from that $k$ is a prime? Please help me.

Answer 1

${F_k}$={o, 1, $x$, $x^2$, ... ,$x^{k-2}$} where $k=p^{m}$, $p$ is a prime number and $m$ is some integer. $p$ is characterstic and $m$ is dimension of the field. $x$ is primitive element if it is the root of the primitive polynomial and $x^{k-1}=1$. Yes, $F_k$ has $k$ elements and $F_p$ is contained inside $F_k$. The notation I have used is power notation which is useful for multiplication operations and ofcourse there is polynomial notation for the field which is useful in addition operations.