If $1, \omega, \omega^2, \ldots \omega^{10}$ are 11 th roots of unity, then product of these roots is 1
If $1, \omega, \omega^2, \ldots \omega^{10}$ are 11 th roots of unity, then product of these roots is 1
$\omega$
-1
1
$\omega^2$
Solution
Given $1, \omega, \omega^2, \ldots, \omega^{10}$ are 11 th roots of unity.
$\Rightarrow 1, \omega, \omega^2, \ldots, \omega^{10}$ are the roots of the equation given below.
or
$
\begin{aligned}
& x=(1)^{1 / 11} \\
or \\
& x^{11}=1 \\
& \Rightarrow \quad x^{11}-1=0 \\
&
\end{aligned}
$
Since, we know that product of roots of a polynomial of degree $n=(-1)^n \cdot \frac{\text { Constant term }}{\text { Coefficient of } x^n}$
$
\therefore 1 \cdot \omega \cdot \omega^2 \ldots \omega^{10}=\frac{(-1)^{11}(-1)}{1}=1
$