# Meaning of $\mathbb{R}[x]$

I ran into this expression in a paper I was reading, and I'm confused about part of the meaning. Here $u$ and $v$ are two polynomials.

$$u, v \in \mathbb{R}[x]$$

I'm not really familiar with usage of $[x]$ here, but if it means "nearest integer", then isn't this expression equivalent to simply:

$$u, v \in \mathbb{Z}$$

• This just means the polynomial ring in the variable $x$ with coefficients in $\mathbb{R}$. – Adrián Barquero Dec 24 '10 at 17:38
• When you write $u \in \mathbb{R}[x]$ it just means that $u = a_n x^n + \dots + a_1 x + a_0$ where the coefficients $a_j \in \mathbb{R}$. – Adrián Barquero Dec 24 '10 at 17:40
• In that context, $[x]$ does not mean nearest integer – J. W. Tanner Jul 10 at 15:24

$\mathbb{R}[x]$ is the set of polynomials (with variable $x$) whose coefficients are taken from $\mathbb{R}$, the set of real numbers. It has got nothing to do with the greatest integer function.

Similarly, people talk about polynomials in $\mathbb{Q}[x]$, where coefficients are rational, or more generally, when the coefficients are taken from an arbitrary Ring.

For instance of a usage:

Definition: A real number $r$ is transcendental if and only if, for every $P \in \mathbb{Q}[x]$, we have that $P(r) \neq 0$.

Generally, if $$\rm\,R \subset S\,$$ are rings and $$\rm\,s\in S\,$$ then $$\rm\,R[s]\,$$ denotes the ring-adjunction of $$\rm\,s\,$$ to $$\rm\,R,\,$$ i.e. the smallest subring of $$\rm\,S\,$$ containing both $$\rm\,R\,$$ and $$\rm\,s\,.\,$$ Equivalently $$\,\rm R[s]$$ is the image of $$\rm\,R[x]\,$$ under the evaluation map $$\rm\,x\mapsto s,\,$$ i.e. elements of $$\rm\,S\,$$ writable as a polynomial in $$\rm\,s\,$$ with coefficents in $$\rm\,R.\,$$
Similarly, if $$\rm\,F \subset E\,$$ are fields and $$\rm\,\alpha\in E\,$$ then $$\rm\,F(e)\,$$ denotes the field-adjunction of $$\rm\,\alpha\,$$ to $$\rm\,F,\,$$ i.e. the smallest subfield of $$\rm\,E\,$$ containing both $$\rm\,F\,$$ and $$\rm\,\alpha.$$
The notation for the polynomial ring $$\rm\,R[x]\,$$ is the special case where $$\rm\,x\,$$ is transcendental over $$\rm\,R\$$ (an "indeterminate" in old-fashioned language),$$\$$ i.e. $$\rm\, x\,$$ isn't a root of any polynomial with coefficients in $$\rm\,R\,$$. One may view $$\rm\,R[x]\,$$ as the most general ring obtained by adjoining to $$\rm\,R\,$$ of a universal (or generic) element $$\rm\,x,\,$$ in the sense that any other adjunction $$\rm\,R[s]\,$$ is a ring-image of $$\rm\,R[x]\,$$ under the evaluation homomomorphism $$\rm\, x\to s\,.\$$
For example, if $$\rm\,R \subset S\,$$ are fields then $$\rm\,R[s]\cong R[x]/(f(x))\,$$ where $$\rm\,f(x)= \,$$ minimal polynomial of $$\rm\,s\,$$ over $$\rm\,R.\,$$ Essentially this serves to faithfully ring-theoretically model $$\rm\,s\,$$ as a "generic" root $$\rm\,x\,$$ of the minimal polynomial $$\rm\,f(x)\,$$ for $$\rm\,s\,.\,$$