The difference between $F(a)$, $F[a]$, $F_q$, and $F_{q^m}$ in ring theory and coding theory

Question

I'm a little confused by some notation in ring theory and coding theory. Assume $F$ is a field. What does the following notation mean (with some examples to understand better)?

• $F(a)$ -- I think this is the smallest field containing $F$ and the root $a$. Is my definition correct??

• $F[a]$?

• $F_q$? (in coding theory)

• $F_{q^m}$

(I know that $F[x]$ is the polynomial ring with coefficients in $F$.)