We know that there are certain definitions and results in mathematics that lead to unintuitive consequences, some mild, which just requires a bit of tweak in perspective, and others straight up mind-boggling. I came across one of the former today, See MathSE. which initially shoke me so much I felt like a truck hit me and had almost sent me down a very badly-timed existential...

Preliminary

First, let’s have some preliminaries to the Primitive Element Theorem.

Separable Extension

We say that the extension $K$ of a field $F$ We usually write $K/F$ for $K$ being an extension of $F$. is separable if it is

• algebraic: i.e. each of its elements is a root of some polynomial of $F[x]$; and
• each $\alpha \in K$ is...

The uncountability of $(0, 1)$ is an important result that demonstrates that $\mathbb{R}$, the set of real numbers, has a greater cardinality, i.e. “bigger size” than that of $\mathbb{N}$, the natural numbers. In other words, the result shows to us that $\mathbb{N}$ is countable while $\mathbb{R}$ is uncountable.

Georg Cantor was the first to have proven this result, and consequently showing to us that there are different “degrees” of infinity....

In PMATH351 F17, I was (re)introduced to the Axiom of Choice. The following is my understanding and interpretation, or at least my attempt at interpreting the axiom, and other statements that are equivalent to it.

Axiom of Choice (AC)

which says that for a family of non-empty sets, the Cartesian product of all such...