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Given a field $K$ (for example $K = \mathbb Q$) and a polynomial $p \in K[X]$ with cyclic Galois group $C_n$ then one only needs to adjoin an $n$th root of some element $k \in K(\zeta_n)$ to $K(\zeta_n)$ to get the splitting field $L=K(\zeta_n,\sqrt[n]{k})$ of $p$. It is possible to actually find the number $k$.

It is said that the roots of $p$ lie in this field $L$, I would like to know how you can find these roots?

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    Given k, you might be able to use: http://math.stackexchange.com/questions/16521/expressing-a-root-of-a-polynomial-as-a-rational-function-of-another-root2011-05-05
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    Also, you can just ask magma. It has reasonably decent factorization routines. An open source implementation is in the RadiRoot package, but I haven't used it much as it focusses on nested radical expressions, whereas I usually want rational polynomials (for instance, in k).2011-05-05

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