Equivalence Class/Examples/Equal Fourth Powers over Complex Numbers
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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 $11.1 \ \text{(a)}$