Let's consider $p$ a prime number. And the primitive p-root of unity $\zeta_p$. In general $G= Gal ( \Bbb Q (\zeta_p),\Bbb Q) \cong \Bbb Z_p^*$ (here we don't need p to be prime). But in the case when $p$ is prime, we know how to find primitive generators for the subfields of $\Bbb Q (\zeta_p)$. If $H\le G$ is a subgroup, then the element $$
\alpha _H = \sum\limits_{\sigma \in H} {\sigma \zeta _p }
$$ is a primitive generator for the fixed field of $H$. Computationally talking, it's convenient to find a cyclic generator of $G$ to have generators for all the subgroup $H$. But when I'm done here, sometimes it's not easy to find minimal polynomials of $\alpha_H$. There is an easy way to do that? Over $\Bbb Q$ first.
For example I have to find all the subfields of $\Bbb Q(\zeta_{11})$ find all the subfields with a primitive element of it , and then find the minimal polynomial over $\Bbb Q$ of all of them.
First note that $\sigma = \sigma_2 $ such that $ \sigma(\zeta_{11}) = \zeta_{11}^2 $ is a generator of $G$ , so the two subgroups of $G$ are: $$
\eqalign{
& H_2 = \left\langle {1,\sigma ^5 } \right\rangle = \left\langle {1,\sigma _{2^5 } } \right\rangle = \left\langle {1,\sigma _{10} } \right\rangle \cr
& H_{5} = \left\langle {1,\sigma ^2 ,\sigma ^4 ,\sigma ^6 ,\sigma ^8 } \right\rangle = \left\langle {1,\sigma _{2^2 } ,\sigma _{2^4 } ,\sigma _{2^6 } ,\sigma _{2^8 } } \right\rangle = \left\langle {1,\sigma _4 ,\sigma _5 ,\sigma _9 ,\sigma _3 } \right\rangle \cr}
$$
Where $ \sigma_a (\zeta_{11}) = \zeta_{11}^a $
Then the two fixed field ( $K_2,K_5$) respectively have primitive elements :
$$
\eqalign{
& \alpha _2 = \zeta _{11} + \zeta _{11} ^{10} \cr
& \alpha _5 = \zeta _{11} + \zeta _{11} ^4 + \zeta _{11} ^5 + \zeta _{11} ^9 + \zeta _{11} ^3 \cr}
$$
And now I have to compute the minimal polynomials over $\Bbb Q$ obviously , I have to consider the relation $ 1+\zeta_{11}+\zeta_{11}^2+….+\zeta_{11}^{10}=0 $ But it's a very complicated and large computation even in this particular case .
My question is , if there exist other ways to compute that miminal polynomial of this elements?
Best Answer
I'm not sure if it answers your question, but the roots of the minimal polynomial are $\sum_{\sigma\in Hg}\sigma\zeta_p$, where $Hg$ runs over all the cosets of $H$ in $G$. You can then find the coefficients of the polynomial simply from Vieta relations. Notice that the coefficients are of the form $m+n(\zeta+\dots+\zeta^{p-1})$ and there you can simply use $\zeta+\dots+\zeta^{p-1}=-1$.