Field of rational functions

Let $𝐷$ be an integral domain and $𝐷 [𝑥]$ be the polynomial ring over $𝐷$ in indeterminate $𝑥$ , which is itself an integral domain The field of rational functions $𝐷 (𝑥)$ in indeterminate $𝑥$ consists of ratios of polynomials in indeterminate $𝑥$ #m/def/ring

𝑓 (𝑥) = 𝑝 (𝑥) 𝑞 (𝑥)

where $𝑝 (𝑥), 𝑞 (𝑥) \in 𝐷 [𝑥]$ , and is the field of fractions

𝐷 (𝑥) = F r a c (𝐷 [𝑥]) = F r a c (F r a c (𝐷) [𝑥])

and a division algebra over $F r a c (𝐷)$ .

Proof

#missing/proof

Properties

Lower bound on the dimension of the field of rational functions

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