Question

Let $1,\omega ,{\omega }^2$ be the cube roots of unity. The least possible degree of a polynomial with real coefficients, having $2\omega ,2+3\omega ,2+3{\omega }^2$ and $2–\omega –{\omega }^2$ as roots is

• A. $8$
• B. $5$
• C. $6$
• D. $4$

Answer 1

The correct option is B $5$

Roots are $2\omega ,2+3\omega ,2+3{\omega }^2,2–\omega –{\omega }^2$.
i.e. $2\omega ,2+3\omega ,2+3{\omega }^2,3$
The roots $2+3\omega ,2+3{\omega }^2$ can come from a 2nd degree polynomial with real coefficients.
Again the root $2\omega$ can come from a 2nd degree polynomial with real coefficients.
The root 3 comes from a linear polynomial.
Thus the least possible degree is 5.