Let u1, u2, …, un be the distinct n th roots of unity and...

Let $u_{1}, u_{2}, \ldots, u_{n}$ be the distinct $n$ th roots of unity and suppose $v$ is a nonzero solution of the equation $$ z^{n}=r(\cos \theta+i \sin \theta) $$ Show that $v u_{1}, v u_{2}, \ldots, v u_{n}$ are $n$ distinct solutions of the equation. [ Remember: Each $\left.u_{i} \text { is a solution of } x^{n}=1 .\right]$

Let $u_{1}, u_{2}, \ldots, u_{n}$ be the distinct $n$ th roots of unity and suppose $v$ is a nonzero solution of the equation
$$
z^{n}=r(\cos \theta+i \sin \theta)
$$
Show that $v u_{1}, v u_{2}, \ldots, v u_{n}$ are $n$ distinct solutions of the equation. [ Remember: Each $\left.u_{i} \text { is a solution of } x^{n}=1 .\right]$

Key Concepts

nth Roots of Unity

This concept refers to the complex solutions of the equation x? = 1, which are evenly spaced on the unit circle in the complex plane. They are pivotal in understanding the symmetries of complex numbers and in generating all solutions to polynomial equations involving roots of unity. Their geometrical distribution results in a natural connection to rotations in the complex plane.

Complex Numbers in Polar Form

Expressing complex numbers in polar form, as r(cos ? + i sin ?) or equivalently re^(i?), is crucial for analyzing their multiplicative behavior. This form encapsulates both the magnitude and the direction (angle) of a complex number, making operations like multiplication straightforward as they translate into scaling the magnitude and adding the angles.

Multiplicative Property of Complex Numbers

When multiplying complex numbers in polar form, their moduli multiply and their angles add. This fundamental property is key in arguments where rotating a complex number by a fixed angle results in another solution of a related equation, as it allows the construction of new solutions by combining known values.

Rotation in the Complex Plane

Multiplying by a nth root of unity has the effect of rotating a complex number by an angle of 2?/n. This geometric interpretation is essential when demonstrating that multiplying a particular solution by all the nth roots of unity produces all distinct rotations (and therefore distinct solutions) of the original complex number.

Uniqueness of Solutions

The argument that distinct nth roots of unity yield distinct products when multiplied by a nonzero complex number relies on the cancellation property in fields. Since each root of unity is unique, their multiplicative action on a nonzero solution does not repeat, ensuring each product is a distinct solution of the complex equation.

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