Foundations of Mathematics

Clock theory

$\textbf{Definition 1.1.}\;$ Define $C_{12}$ to be hours on a clock. In fact we can define clocks of any number $n$ of hours which we denote $C_n$. The canonical notation for elements in $C_n$ is given as follows

$$ \begin{align} C_n := \{[0:00],[1:00],...,[(n-1):00]\}\end{align}. $$

We can also add two binary operators to make a clock a ring. These are given by

$$ \begin{align} + : C_n \times C_n &\rightarrow C_n \\ ([a:00], [b:00]) &\mapsto \text{$b$ hours after $a$ o'clock} \\\\ \times : C_n \times C_n &\rightarrow C_n \\ ([a:00], [b:00]) &\mapsto \text{$a \times b$ hours after midnight} \end{align} $$

$\textbf{Exercise 1.2.}\;$ Show that $C_n$ with these operators is actually a ring.

$\textbf{Definition 1.3.}\;$ The $\textit{eternal clock}$ $C$ is defined to be the clock with countably many hours as well as a notion of negative hours. We can give the eternal clock a ring structure as well by giving it the same $+$ and $\times$ operators defined above. i.e. we have \[ C := \{ [a:00] \;|\; a \in \mathbb{Z}. \} \]

$\textbf{Theorem 1.4.}\;$ Every finitely generated abelian group can be expressed as the direct product of finitely many clocks and eternal clocks.

$\textbf{Theorem 1.5.}\;$ The field extension $ \mathbb{Q}(\zeta_n) / \mathbb{Q} $ where $\zeta_n$ is a primitive $n$th root of unity with $n$ an odd prime has Galois group given by a clock.

Tennis court theory

Tennis court theory is concerned with finding the mysterious $\textit{tennis court homomorphism}$ to help prove the solvability of the symmetric group $S_4$ and the existence of a formula with radicals for the quartic polynomial. Finding this tennis court homomorphism gives us the following abliean series of $S_4$ \[ S_4 \unrhd A_4 \unrhd T \unrhd \{1\}, \] where $T$ is the kernel of the tennis court homomorphism and hence is a normal subgroup.

Representation theory of finite groups

Awesome but not as cool as Rubik's cubes.