Field Automorphisms: A Nice Touch on "Ambiguity"

I am sure that the title of the post is going to confuse many readers, but for lack of a better title please be content with it. Rest assured that the post itself is quite unambiguous. We would like to throw some light on the a special kind of ambiguity which we find in mathematical systems (here mainly in algebra).

A Taste of Modern Algebra: Remainder Theorem for Polynomials

Introduction

In high school curriculum we are taught the "Remainder Theorem" as one of the important results of Algebra. The theorem and its proof are quite simple, but the practical application and theoretical ramifications of this result are quite interesting. To recall, the theorem is stated as follows:

Remainder Theorem: If $f(x) = a_{0}x^{n} + a_{1}x^{n - 1} + \cdots + a_{n - 1}x + a_{n}$ is a polynomial with real coefficients then the remainder obtained on dividing $f(x)$ by $(x - a)$ is $f(a)$.