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Thanks for your answer Ram. But I will change my question , because I want a more directly answer. And I'll write exactly what I want.

Edited question:

Let's consider a finite Galois extension $L/K$. And let's take $\alpha \in L$. Let $ G = Gal(L,K)$ . Let's consider the conjugates of $\alpha$ i.e $ \sigma(\alpha) $ where $\sigma \in G $. Let's call the different roots of the minimal polynomial of $\alpha$ by $m_{\alpha}(x) \in K[x] $ by $ \alpha_1 , ... \alpha_r $. Since $L/K$ is Galois, we know that these roots are contained in the set of all the conjugates of $\alpha$.

Let $ d= [K(\alpha):K]$ . What I want to prove is that each $\alpha_i $ appears exacly $d$ times in the conjugates.

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    @KCd, the question w$a$s as such that it wants to $b$e comfortable with the whole 11 pages :-).2012-11-16

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