# Difference between $F[x]$ and $F(x)$

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.

• $F[x]$ and $F(x)$ both contain copies of $F$ as the constant elements; $F[x]$ as a subring, $F(x)$ as a subfield. Mar 20, 2015 at 5:14

You are right about the definitions. An element of $\mathbb R(x)$ not in $\mathbb R$ is for example $3x$. $3$ would be in $\mathbb R$ and $\mathbb R(x)$ at the same time.
In general, $F(x)$ contains $F[x]$ which in turn contains $F$.
• Can you explain why $F(x)$ contains $F[x]$? I am writing $F(x)=\{a+bx : a,b \in R\}$ so since $x$ is not in $F$ I think that $x^2$ is not in $F(x)$ for example. Mar 21, 2017 at 14:20
• @Ninja $x^2$ is a rational function, so it is in F(x). Oct 7, 2021 at 1:04