Why does any extension of $\mathbb{Q}$ that contains a root of $x^6+3$ have to contain a primitive $6^{th}$ root of unity? The roots of this equation are $\{\sqrt[6]{3}e^{\pi i/2},\sqrt[6]{3}e^{5\pi i/6},\sqrt[6]{3}e^{7\pi i/6},\sqrt[6]{3}e^{3\pi i/2},\sqrt[6]{3}e^{11\pi i/6},e^{13\pi i/6}\}$. A primitive $6^{th}$ root of unity has the form $\frac{1}{2}+\frac{\sqrt{3}}{2}i$…
I don't see how to get $\frac{1}{2}+\frac{\sqrt{3}}{2}i$ from $\sqrt[6]{3}e^{\pi i/2}$, for example.
Best Answer
Let $\alpha$ be such a root, then $\beta=\alpha^3$ satisfies $\beta^2+3=0$, hence ${1+\beta\over 2}={1+\alpha^3\over 2}$ is a $6^{th}$ root of $1$.
You can even verify this directly:
Now use that $\beta^2=-3$ to reduce this to:
$$(1+5\cdot(-3)+15\cdot(-3)^2+(-3)^3)+\beta(6+20\cdot(-3)+6\cdot(-3)^2)$$ $$=(1-45+135-27)+\beta(6-60+54)= 64+0\beta $$ $$= 2^6.$$
And this is primitive as if we only cube things we get