Notation wise, what is the difference between $F[x]$ and $F(x)$? Is $F[x]$ the ring of polynomials with coefficients in $F$, and $F(x)$ the field of rational functions with coefficients in $F$?
I am asking because I am trying to determine if this statement is true or false:
An element of the field $F(x)$ of rational functions is transcendental over $F$ if and only if it is not in $F$
and I'm not sure what an element of $F(x)$ not being in $F$ means.
Thanks for your help.