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Let $G$ be the Galois group of an irreducible polynomials $f(x)$ in $\mathbb{Q}[x]$. Let $K$ be the splitting field of $f(x)$.

From the fundamental theorem of Galois theory we have that the intermediate fields between $K$ and $\mathbb{Q}$ are in bijective correspondence with the subgroups of $G$.

Is there any way we can get the polynomials whose Galois groups are the subgroups of $G$, given $f(x)$ ?

To put it in an other way is there any relation between $f(x)$ and the polynomials corresponding to the normal extensions of $\mathbb{Q}$ contained in $K$ ?

By polynomial corresponding to normal extensions I mean, given a normal extension $E$ of $\mathbb{Q}$ contained in $K$, The polynomial $p(x)$ whose splitting field is $E$.

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    @ArturoMagidin Now I remember Abel saying a similar kind of thing about *asking* a question. :)2011-10-22

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I'm going to (try to) answer the question the way I phrased it in my first comment. It depends a bit on just what one is given. I'll take the easy way out and assume one is given a basis $\lbrace\alpha_1,\dots,\alpha_r\rbrace$ for $E$ over $\bf Q$ (I might also assume given a basis for $K$ over $\bf Q$, and the actions of the elements of $G$ on that basis, and which elements of $G$ are in the subgroup corresponding to the field $E$). Then with a little experimentation one can find $\beta=c_1\alpha_1+\cdots+c_r\alpha_r$ with $c_i$ integers such that $E={\bf Q}(\beta)$. Then you can find all the conjugates of $\beta$ over $\bf Q$ and use that to find the minimal polynomial $p$ for $\beta$ over $\bf Q$. And $p$ is what you want.

If what you really want is some way of just staring at $f$ long enough and hard enough for $p$ to appear, I suspect you're asking for too much.

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    I just wanted to know what can be said in a more general situation (a mathematical habit inherently in me from years, don't know if it is good/bad) than the $p$-extensions, nevertheless I shall ask a question about it separately sooner or later. Not much seems to be working well this question. Thanks for your answer and patient comments and criticism.2011-10-23