Definition for set of ratios of polynomials

Just as there is a difference between polynomials in a variable $x$ and polynomial functions in the variable $x$, I am looking for the terminology for the ratio of two polynomials. They "look" like rational functions, but there is no regard to domains and ranges.

For example, \begin{equation*} f(x) = \frac{6x + 9}{3x - 12} \qquad \text{and} \qquad g(x) = \frac{2x + 3}{x - 4} \end{equation*} are the same Mobius transformation, but they have different coefficients and constant terms.

Is there an algebraic term for the ratio of polynomials in the variable $x$, and is there an algebraic term for the ring of all such ratios?

1 Answer

The field of all quotients of polynomials in the variable $X$ over a field $K$ is most often denoted by $$K(X)= \{\frac{f}{g} | f,g\in K[X], g \neq 0\}$$ and is called the function field. The elements are called rational fraction or also rational function. Still, the quotients are really just regarded as fractions of polynomials and not as functions. It is the field of fractions of the integral domain $K[X]$, that is the smallest field that contains all polynomials.

• You have reminded me of this phrase. $\mathbb{R}(x)$ is the field of fractions of the polynomial ring $\mathbb{R}[x]$ over the field of real numbers." – user232552 Jan 19 '18 at 18:21
• In this field, are the quotients $f(x)$ and $g(x)$ that I have in my post distinct elements? – user232552 Jan 19 '18 at 18:21
• @Adelyn No, they are not. As in $\mathbb{Q}$, in the construction of $K(X)$ one identifies fractions if they have the same reduced presentation, that is $\frac{f}{g}=\frac{f'}{g'}$ if and only if $fg'=f'g$. – Verdruss Jan 19 '18 at 18:28