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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.

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    $\begingroup$ $F[x]$ and $F(x)$ both contain copies of $F$ as the constant elements; $F[x]$ as a subring, $F(x)$ as a subfield. $\endgroup$
    – Gyu Eun Lee
    Mar 20, 2015 at 5:14

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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 $F$.

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  • $\begingroup$ But 3x would also be in R[x], yes? So R(x) should be a bigger field than R[x]. $\endgroup$
    – jstnchng
    Mar 20, 2015 at 5:17
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    $\begingroup$ Please see edit. $\endgroup$
    – please delete me
    Mar 20, 2015 at 5:18
  • $\begingroup$ Thanks for the clarification Jasper. $\endgroup$
    – jstnchng
    Mar 20, 2015 at 5:22
  • $\begingroup$ 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. $\endgroup$
    – Ninja
    Mar 21, 2017 at 14:20
  • $\begingroup$ @Ninja $x^2$ is a rational function, so it is in F(x). $\endgroup$
    – cineel
    Oct 7, 2021 at 1:04

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