Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

up vote 4 down vote accepted

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 $\zeta+\dots+\zeta^{p-1}=-1$.

share|improve this answer
1  
Thanks man! With your answer I have a factorization of the minimal polynomial , so it only left to multiply :D!! –  Daniel Nov 4 '12 at 20:11

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.