If 1, α1, α2, ……, αn-1 are the nth roots of unity and if ωis a non real 5 th root of unity then

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If 1, α1, α2, ……, αn-1 are the nth roots of unity and if ωis...

If $1, \alpha_{1}, \alpha_{2}, \ldots \ldots, \alpha_{\mathrm{n}-1}$ are the nth roots of unity and if $\omega$ is a non real 5 th root of unity then $\left(\omega-\alpha_{1}\right)\left(\omega-\alpha_{2}\right) \ldots \ldots$ $\left(\omega-\alpha_{n-1}\right)$ is not non-real provided $n$ is of the form (a) $5 \mathrm{~K}+2$ (b) $5 \mathrm{~K}+1$ (c) $5 \mathrm{~K}$ (d) $5 \mathrm{~K}+3$

If $1, \alpha_{1}, \alpha_{2}, \ldots \ldots, \alpha_{\mathrm{n}-1}$ are the nth roots of unity and if $\omega$ is a non real 5 th root of unity then $\left(\omega-\alpha_{1}\right)\left(\omega-\alpha_{2}\right) \ldots \ldots$ $\left(\omega-\alpha_{n-1}\right)$ is not non-real provided $n$ is of the form
(a) $5 \mathrm{~K}+2$
(b) $5 \mathrm{~K}+1$
(c) $5 \mathrm{~K}$
(d) $5 \mathrm{~K}+3$

Key Concepts

Real Values from Complex Sums

The expression (?? – 1)/(? – 1) represents a sum of successive powers of a non?real 5th root of unity. Although ? is non?real, for certain values of n (specifically when the number of terms in the sum corresponds to a complete number of cycles or one extra term) the sum becomes real. This happens because the symmetry of the roots over the circle causes the imaginary parts to cancel in those special cases.

Modular Arithmetic in Exponents

Because ? is a root of unity (and in this specific situation a 5th root), its powers are periodic with period 5; that is, ?? = 1. This means that when computing sums like 1 + ? + ... + ?^(n–1), one only needs to know n modulo 5. This modular arithmetic plays a key role in determining the nature (for example, whether the result is real or not) of expressions involving these sums.

nth Roots of Unity

The nth roots of unity are the complex solutions of the equation x? = 1. They are evenly spaced on the unit circle in the complex plane and have many symmetrical properties. When dealing with products or sums involving these roots, one often exploits the fact that they are distributed uniformly, which can simplify expressions and lead to results that depend only on the modulus of certain parameters (such as n modulo some number).

Geometric Series Summation

When one has a sum of terms of the form 1 + ? + ?² + … + ?^(n–1) where ? is a complex number (particularly a root of unity), the sum is a finite geometric series. Its closed?form expression is (?? – 1)/(? – 1) when ? ? 1. Using this formula is a standard technique to simplify the product or sum over roots, as it connects the periodicity of the powers of ? with modular conditions on n.

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