What does the word rational in rational fraction refer to

Question

I understand that a rational fraction is an element in the fraction field $F(X)$ and that a fraction $f$ is represented by a quotient of two poloynomials $A\over B$ under the relation ${A\over B}={C\over D}\leftrightarrow AD=BC$, but what I don't understand is the word rational in this context what does it refer to ? why not simply say a fraction , or more preciseley a fraction of polynomials ? what's the meaning of rational here especially when the field $F$ is not the field $\mathbb Q$ of rationals !! thank you for your help !

Answer 1

This probably has some deeper historical reasons behind it. But in any case, we need to be able to make a distinction between a fraction over $F$: $$\frac{a}{b}, \qquad a,b,\in F,$$ and a rational fraction over $F$: $$\frac{p(X)}{q(X)},$$ where $p,q$ are polynomials in $X$ with coefficients in the field $F$. If you also call the latter a fraction (which it is, but not over $F$), there is a possibility for confusion. And although fraction of polynomials would be appropriate, it is longer (and that's where the historical reasons probably kick in).

Finally, let's not forget that the word rational refers to a ratio; rational numbers are ratios of integers, and rational fractions are ratios of polynomials.