\text{ and } \\
 \tau(x+iy) &  = &  -x+iy.
 \end{aligned}$$ Thus, $\sigma$ is the complex conjugation and $\tau$ is like a real brother of $\sigma$. Note that if we apply $\sigma$ or $\tau$ twice, we get the identity map: $\sigma^{2} = 1 = \tau^{2}$. Also, we see that $\sigma \tau = \tau \sigma = - 1$. Geometrically, the maps $\sigma \tau$ and $\tau \sigma$ are rotations by $180$ degrees in the complex plane. The set of maps $$\{1, \sigma, \tau, \sigma \tau\}$$ forms a group under composition and is called the Klein four group or just Klein group, often denoted by $V_{4}$. One would guess that the letter $V$ here is a sign of victory but the reason is that “*Vier*” in German means “four.” “Klein” in German also means “small” and indeed Klein group $V_{4}$ having only four elements is quite small. It is an absolutely amazing fact that this small and ostensibly innocent group contains remarkable richness and that important mathematics can be developed by just studying this one group. The world’s smallest field is ${\mathbb{F}_2}= \{ 0, 1\}$, and one can think of this as a toy model of complex numbers. The problem to be investigated in 