Question About Notation In Field Theory $F(x)$ vs. $F[x]$

Question

I have a question about notation specifically square brackets $[$ and round brackets $($.

My textbook doesn't explain any of this and I cannot find a reliable source online to confirm the difference.

So my question is: What is the difference between round brackets and square brackets in terms of notation in Field Theory? For example, I see $F(x)$ and $F[x]$ in my textbook and I've always assumed they were the same thing. But apparently they're not. Is there ever a time they're the same?

I wanted to know the difference, it may be a silly question but it's something I want to make sure I understand.

Answer 1

Consider $x$ as an element contained in some extension of $F$; then $F(x)$ is the smallest field that contains both $F$ and $x$. On the contrary $F[x]$ is the smallest ring that contains both $F$ and $x$. Clearly $F[x]\subseteq F(x)$, and in some cases they are equal. You can prove easily (or see it on Morandi's book) that

$$F[x]=\{f(x):\textrm{$f$ is a polynomial with coefficients in $F$}\}$$ $$F(x)=\left\{\frac{f(x)}{g(x)}:\textrm{$f,g$ are polynomials with coefficients in $F$ and $g(x)\neq0$}\right\}$$

Answer 2

$F[x]$ denotes the ring of (formal!) polynomials over $F$ in the indeterminate $x.$
$F(x)$ denotes its fraction field - the so-called rational "functions" in $x.$

The same notation is used for adjunctions of elements to rings and fields, i.e. if $\,\alpha\,$ is an element of some extension ring $\,E\,$ then $\,F[\alpha]\,$ denotes the smallest subring $\,R\,$ of $\,E\,$ that contains $\,F\,$ and $\,\alpha.\,$ Equivalently, it is the image of $\,F[x]\,$ under the evaluation map $\,x\mapsto \alpha,\,$ i.e. the set of all elements expressible as polynomials in $\,\alpha\,$ with coefficients in $\,F.\,$ If $\,\alpha\,$ is transcendental (= not algebraic) over $\,F,\,$ i.e. $\,\alpha\,$is not a root of any nonzero polynomial $\,f\in F[x],\,$ then there is a ring isomorphism $\,F[\alpha]\cong F[x].\,$ Thus the polynomial ring $\,F[x]\,$ can be viewed as the ring obtained by adjoining to $\,F\,$ any element that is transcendental over $\,F,\,$ e.g. $\,\Bbb Q[x]\cong \Bbb Q[\pi].\,$ Similarly for the case of fields.