Find all subfields of the splitting field of $x^{12}-1$.
My work so far:
We need to find the subfields of $\mathbb{Q}(\zeta_{12})$. We know the Galois group over $\mathbb{Q}$ is the 4-klein group so there are 3 subfields. I found 2, just by guessing $\mathbb{Q}(\sqrt{3}i)$ and $\mathbb{Q}(i)$ but I cannot find the last subfield. The only way I could think of finding it, is to find the minimal polynomial so that i know what the algebraic relations are between $1,\zeta_{12},\zeta_{12}^2,\zeta_{12}^3$, and then find the fixed field of each automorphism. However trying that i realized how obnoxious that is, and computationally heavy. How do I find the last field?
Best Answer
The field you missed is the real subfield of $\mathbb Q(\zeta_{12})$, which is $\mathbb Q(\sqrt3)$.
If $\zeta$ is a complex root of $x^n-1$ with $n>2$, then $\zeta^{-1}$ is the complex conjugate of $\zeta$,
so $\mathbb Q(\zeta+\zeta^{-1})$ is the real subfield of the complex field $\mathbb Q(\zeta)$.