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Let $\varphi:K_1\to K_2$ be an isomorphism of fields. Let's consider a polynomial $p(x)\in K_1[x]$, and it's splitting field extension $ E_1/K_1 $, also the polynomial $\varphi(p(x))$ $\in K_2[x]$, with it's corresponding splitting field extension $E_2/K_2$. I want to know in how many ways I can extend the isomorphism $\varphi$ into an isomorphism $\sigma:E_1\to E_2$, I know that the degree of this is bounded by $[E:K]$, and it's and equality, at least in the case that $p(x)$ it's a separable polynomial. I want to know if the number of extensions it's exactly the number of distinct roots of $p(x)$.

Where the polynomial $\varphi(f)$ means the following: If $p=\sum a_ix^i $ then $\varphi(f)=\sum \varphi(a_i)x^i $

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Certainly not. If $p$ splits in $K$ then there is only one "extension", independent of the number of roots of $p$. Do you want $p$ to be irreducible? –  Tom Bachmann Sep 24 '12 at 17:53
    
And the irreducible case should follow in general by the observation that any extension is reachable by a sequence of simple extensions, and induction. –  Tom Bachmann Sep 24 '12 at 18:10

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