The discussion centers on the utility of rational functions in one indeterminate, specifically their role as quotients of polynomials. Participants highlight their relevance in algebraic structures, such as quotient fields of integral domains and transcendental field extensions. Key applications include Möbius transformations in complex analysis and their significance in algebraic geometry for describing local behaviors of curves. The conversation concludes that while rational functions may seem trivial, they serve as essential examples in abstract algebra, particularly in understanding the structure of fields and rings.
PREREQUISITESMathematicians, algebraists, and students studying abstract algebra, particularly those interested in the applications of rational functions and their role in algebraic structures.
Kind of strange, to me at least, to see how Complex lines become spheres through projectivization.mathwonk said:my bad, mixing my metaphors. If I am going to speak of the projective line as the [riemann] sphere, then i should speak of a [complex curve] as a riemann surface. thanks for this clarification.