Question
Show that the roots of the equation 1 0 z4 − = form a cyclic group of order 4.
Answer :
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Skip in 10
To solve the equation \( z^4 = 1 \), we want to find the complex roots of this equation. The roots of the equation \( z^4 = 1 \) are the fourth roots of unity, and they can be expressed in the form: \[ z = e^{2\pi i n / 4} \quad \text{for} \quad n = 0, 1, 2, 3 \] This gives the following four roots: 1. \( z_0 = e^{2\pi i (0)/4} = e^{0} = 1 \) 2. \( z_1 = e^{2\pi i (1)/4} = e^{i\pi/2} = i \) 3. \( z_2 = e^{2\pi i (2)/4} = e^{i\pi} = -1 \) 4. \( z_3 = e^{2\pi i (3)/4} = e^{i3\pi/2} = -i \) Now, we need to show that these roots form a cyclic group of order 4 under multiplication. ### Step 1: Verify the Group Properties A set of elements
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