TITLE: Minimum polynomial of a root involving the 7th root of unity
QUESTION [6 upvotes]: Let ω be a primitive 7th root of unity in $\Bbb C$ and set $α := ω + ω
^6$
. Determine, with
justification, the minimum polynomial of α over Q.
Would one use logs in such a question , or how should one begin? it's the roots of unity that are throwing me off, I know how to these minimal polynomial questions for radicals etc..

REPLY [2 votes]: It's important to see the easy reason you expect a cubic, since that also gives you a direct way to compute the polynomial.  The Galois group for $\omega$ is $\mathbb{Z}/6\Bbb{Z}$, generated by sending $\omega$ to $\omega^{3}$.  Applying this once, $\omega^{3} + \omega^{-3}$ is a conjugate of $\omega + \omega^{-1}$.  Applying it again yields $\omega^{2} + \omega^{-2}$ since 3*3 = 2 mod 7.  Applying a third time gets you back where you started since 3*3*3 = 27 = -1 mod 7.  Thus, the minimal polynomial must be $(x - (\omega + \omega^{-1})) + (x - (\omega^{2} + \omega^{-2})) + (x - (\omega^{3} + \omega^{-3}))$, and the rest is simple arithmetic (and remembering the all 7 7th roots of 1 sum to 0, so that the six of them other than 1 sum to -1).