Question

Consider the following statements:
(1) ${({\omega }^{10}+1)}^7+\omega =0$
(2) ${({\omega }^{105}+1)}^{10}+\omega =p^{10}$ for some prime number $p$ where $\omega \ne 1$ is a cubic root of unity.
Which of the above statements is/are correct?

• A. (1) only
• B. (2) only
• C. Both (1) an (2)
• D. Neither (1) nor (2)

Answer 1

Answer: (D)

Neither (1) nor (2)

By the property of cube root of unity , we have ${\omega }^3=1$ and $1+\omega +{\omega }^2=0$

Now ${\omega }^{10}={\omega }^9\omega ={{(\omega }^3)}^3\omega =1\cdot \omega =\omega$

$\therefore {{(\omega }^{10}+1)}^7+\omega$

$={(\omega +1)}^7+\omega$

$={(-{\omega }^2)}^7+\omega$

$=-{\omega }^{14}+\omega$

$=-{\omega }^{12}{\omega }^2+\omega$

$={-\omega }^2+\omega \ne 0$

Also ${{(\omega }^{105}+1)}^{10}+\omega$

$={(({\omega }^3{)}^{35}+1)}^{10}+\omega$

$={(1+1)}^{10}+\omega$

$={\left(2\right)}^{10}+\omega$ , which is not equal to some power of a prime number.

Hence, none of the statements are true.