Equivalence Class/Examples/Equal Fourth Powers over Complex Numbers

Example of Equivalence Class

Let $\C$ denote the set of complex numbers.

Let $\RR$ denote the equivalence relation on $\C$ defined as:

$\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$

Then the equivalence class of $1 + i \sqrt 3$ under $\RR$ is:

$\eqclass {1 + i \sqrt 3} \RR = \set {1 + i \sqrt 3, -1 - i \sqrt 3, -\sqrt 3 + i, \sqrt 3 - i}$

Proof

From Equivalence Relation Examples: Equal Fourth Powers over Complex Numbers, $\RR$ is an equivalence relation.

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We have that:

$1 + i \sqrt 3 = 2 \cis \dfrac \pi 3$

Hence $\paren {1 + i \sqrt 3}^4 = 16 \cis \dfrac {4 \pi} 3$

So:

$\eqclass {1 + i \sqrt 3} \RR = \set {2 \map \cis {\dfrac \pi 3 + \dfrac {n \pi} 2}: n \in \set {0, 1, 2, 3} }$

Hence the result.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.1 \ \text{(a)}$