Help with notation (primality testing and factorization course)

Question

What are the terms $\mathbb{F}_p$, $\mathbb{F}_q$and $\mathbb{F}[x]$? I know it has something to do with fields but not really sure

Answer 1

$\Bbb F_p$ refers to the field of $p$ elements, where $p$ is some prime. There is only one such field, which is also $\Bbb Z_p$. I.e., it can be identified with the set $\{0, 1,..., p-1\}$ with addition and multiplication modulo $p$. $\Bbb F_q$ is of course the same, except the prime defining the field is labeled $q$ instead of $p$.

In $\Bbb F[x], \Bbb F$ represents some arbitrary field. For instance $\Bbb F$ could be $\Bbb R, \Bbb Q, \Bbb C, \Bbb F_2$, etc. $\Bbb F[x]$ is then the ring of all polynomials in the variable $x$ with coefficients in $\Bbb F$. Note that $\Bbb F[x]$ is not field, as only the non-zero constants have inverses. However, the notation $\Bbb F(x)$ denotes the set of rational functions in $x$ over $\Bbb F$, which is a field.