Power Function on Complex Numbers is Epimorphism/Examples/Cube

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Example of Use of Power Function on Complex Numbers is Epimorphism

Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.

Let $f_3: \C_{\ne 0} \to \C_{\ne 0}$ be the mapping from the set of complex numbers less zero to itself defined as:

$\forall z \in \C_{\ne 0}: \map {f_n} z = z^3$

Then $f_3: \struct {\C_{\ne 0}, \times} \to \struct {\C_{\ne 0}, \times}$ is a group epimorphism.

The kernel $U_3$ of $f_3$ is the set of complex $3$rd roots of unity:

$U_3 = \set {1, \omega, \omega^2}$


\(\ds \omega\) \(:=\) \(\ds \cos \dfrac {2 \pi} 3 + i \sin \dfrac {2 \pi} 3\)
\(\ds \omega^2\) \(:=\) \(\ds \cos \dfrac {4 \pi} 3 + i \sin \dfrac {4 \pi} 3\)

Hence for all $ z \in \C_{\ne 0}$, the coset $z U_3$ is the set:

$z U_3 = \set {z, z \omega, z \omega^2}$

and multiplication on $\C_{\ne 0} / U_3$ of all such cosets satisfies:

$\set {z_1, z_1 \omega, z_1 \omega^2} \times \set {z_2, z_2 \omega, z_2 \omega^2} = \set {z_1 z_2, z_1 z_2 \omega, z_1 z_2 \omega^2}$

Hence the associated isomorphism $g: \C_{\ne 0} / U_3 \to \C_{\ne 0}$ takes the equivalence class:

$\set {z, z \omega, z \omega^2}$

into the cube:

$z^3 = \paren {z \omega}^3 = \paren {z \omega^2}^3$

of any one of its elements.