Polynomial ring

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 ring

𝑓(𝑥)=𝑝(𝑥)𝑞(𝑥)

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

𝐷(𝑥)=Frac⁡(𝐷[𝑥])=Frac⁡(Frac⁡(𝐷)[𝑥])

and a division algebra over Frac⁡(𝐷).

Proof

proof

Properties

• Lower bound on the dimension of the field of rational functions

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