 # Let X be some affine algebraic variety over <mrow class="MJX-TeXAtom-ORD"> <mi mathva Roland Waters 2022-06-25 Answered
Let $X$ be some affine algebraic variety over $\mathbb{k}$ (i.e. some closed subset in ${\mathbb{A}}_{\mathbb{k}}^{n}$). First suppose $X$ to be irreducible. Then the algebra $\mathbb{k}\left[X\right]$ is a domain and we can consider the field of rational functions ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_{\mathbb{k}\left[X\right]}=\mathbb{k}\left(X\right)$. Could you explain me how to build an analogue of this field in the case when $X$ is not necessarily irreducible? Then $\mathbb{k}\left[X\right]$ must not be a domain and we are to build some kind of localization?
Also, what is the destination of rational functions? Why we cannot be satisfied with only regular maps and regular functions?
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The analogue of the quotient field for a ring with zero divisors is the total ring of fractions: basically, just invert everything that is not a zero divisor. Geometrically, an element of this ring can be viewed as a collection of rational functions, one on each irreducible component of $X$, such that they coincide on intersections.
Rational functions are important for a wide variety of reasons. Asking this question is like asking why $\mathbf{Q}$ is important. Why weren't we happy with $\mathbf{Z}$? Well, it wasn't big enough for what we wanted to do, so we enlarged it.

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