On the field of definition of a cubic rational function and its critical points

TL;DR

This paper proves that cubic rational functions over complex numbers with real critical points are equivalent to real functions, but this property does not extend to p-adic fields, highlighting differences in algebraic structures.

Contribution

The paper provides an algebraic proof linking real critical points to real rational functions over complex numbers and demonstrates the failure of this property over all p-adic fields.

Findings

Cubic rational functions over ℂ with real critical points are equivalent to real functions.

The generalization to ℚ_p fields fails for all primes p.

The proof relies solely on algebraic methods.

Abstract

Using essentially only algebra, we give a proof that a cubic rational function over $\mathbb{C}$ with real critical points is equivalent to a real rational function. We also show that the natural generalization to $\mathbb{Q}_p$ fails for all $p$.