Let \(K\) be a field. The field of rational functions in the variable \(x\) over \(K\) is the field of fractions of the polynomial ring \(K[x]\): \[ K(x)\coloneqq\theset{\frac{P(x)}{Q(x)} \st P,Q \in K[x],\ Q\neq 0}. \]
\(\CC(x)\) is the field of meromorphic functions on the Riemann sphere \(\hat\CC=\CC\cup\theset{\infty}\). Every rational function is meromorphic on \(\hat\CC\). Conversely, a meromorphic function \(h\) on the compact surface \(\hat\CC\) has finitely many poles; subtracting from \(h\) its principal parts at the poles in \(\CC\), which are rational functions, and the polynomial principal part at \(\infty\) leaves a holomorphic function on \(\hat\CC\), which is constant by Liouville’s theorem, so \(h\in\CC(x)\).
What is the field of rational functions $K(x)$?
What are the elements of the field of rational functions $K(x)$?