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Question

Let 1,w,w2 be the cube root of unity. The least possible degree of a polynomial with real coefficients having roots 2w,(2+3w),(2+3w2),(2ww2) is ____.

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Solution

Since 1,ω,ω2 are cube roots of unity, the following three properties hold:
a) ω3=1
b) ω2+ω=1
c) ω,ω2 are conjugates to each other
Since ω,ω2 are conjugates to each other a+bω,a+bω2 are also conjugates a,bϵR and b0 because a is only added to real part after multiplying with b which doesn't affect its properties.
Thus, 2+3ω and 2+3ω2 are conjugates.
Also, 2ωω2=3 (substituting for ω2ω=1)
Moreover, for any polynomial equation with real coefficients, complex roots exists in pairs. Since we already have a pair of complex roots and a real root, the conjugate of the root 2ω, which is 2ω2, is sufficient to make all of them, the roots of a polynomial.
Thus, with the roots 2ω,2ω2,(2+3ω),(2+3ω2),(2ωω2), minimum degree of the required polynomial is 5

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